Courses

The tangents at two points $A$, $A'$ of a circle $S$ meet in $T$. The mid-points of $TA$, $TA'$ are ...

$OA$, $OB$, $OC$ are three lines through the point $O$. The angles $BOC$, $COA$ and $AOB$ are, respe...

Points $X$ and $Y$ are chosen, on the perpendiculars (produced if necessary) from the vertices $A$ a...

$ABC$ is an isosceles triangle, with $AB = AC$, $I$ is the centre of the inscribed circle. $S, I_1$ ...

A cube stands on a horizontal surface, and supports a second cube of equal size which is balanced on...

$ABC$ is a non-isosceles triangle, with $M$ the mid-point of $BC$. A line passes through $A$, $B$ in...

A straight line meets the sides $BC$, $CA$, $AB$ of a triangle $ABC$ in $L$, $M$, $N$ respectively. ...

A fixed point $K$ lies inside a triangle $ABC$ and a circle through $A$ and $K$ meets $AB$, $AC$ aga...

A regular dodecahedron is bounded by twelve regular pentagons. Find to the nearest degree the obtuse...

$A$, $B$, $C$, $D$, $E$, $F$, $G$ are consecutive vertices of a regular polygon of $n$ sides ($n \ge...

A square $ABCD$ is such that $A$ lies on $y = 0$, $C$ on $x = 0$, while $B$ and $D$ lie on the circl...

$ABCDE$ is a regular pentagon of side 1. $BD$ and $CE$ meet in $A'$, and $DA$ and $BC$ meet in $C'$....

Each of three circles $C_1$, $C_2$ and $C_3$ meets the other two, but they do not have a common inte...

P and Q are two points on a semi-circle whose diameter is AB; AP and BQ meet in M, AQ and BP meet in...

Find the locus of a point P which moves in a plane containing three distinct fixed points $A_1$, $A_...

$ABC$ is a triangle, and $BCA', CAB', ABC'$ are equilateral triangles; $A, A'$ being on opposite sid...

The tangents at points $A$ and $B$ of a circle $\Gamma$ meet at a point $O$. A chord of $\Gamma$ pas...

A solid is constructed by cutting the corners off a cube in such a way that its set of faces consist...

Let $C_1$, $C_2$ and $C_3$ be circles in the plane, each pair of which intersect in two points. The ...

A triangle $ABC$ has area $\Delta$, and $P$ is an interior point. The line through $P$ parallel to $...

Show that it is not always possible to inscribe a circle within a convex quadrilateral with sides (t...

Given a triangle $ABC$ show that it is possible to construct three mutually touching circles with ce...

The great grey green greasy Limpopo river is 1 kilometre wide and flows with negligible speed betwee...

Show that angles subtended by a chord of a circle at the circumference and in the same segment are e...

Let $ABCDE$ be a regular pentagon and let $AC$ and $BE$ intersect at $H$. Prove that $AB = CH = EH$ ...

Prove that the three altitudes (i.e. perpendiculars from the vertices to the opposite sides) of a tr...

Five points $A$, $B$, $C$, $D$ and $E$ lie in that order on a circle. The lengths $AB$ and $DE$ are ...

Let a convex quadrilateral $Q$ have sides $a$, $b$, $c$, $d$. Let $a$ and $b$ include the angle $\al...

\begin{enumerate}[label=(\alph*)] \item Prove that the angle subtended by a chord of a circle at any...

A triangular lamina is given, and instruments capable of measuring lengths and angles to within know...

A leaf of a book is of width $a$ and height $b$, where $3a \leq 2\sqrt{2}b$; the lower corner of the...

A running track is in the form of a convex circuit. The width of the track is $d$. By how much does ...

$ABC$ is an acute angled triangle and $P$ is the foot of the perpendicular from $A$ to $BC$. $X$ is ...

A model of hyperbolic (non-Euclidean) geometry is given as follows. The points (called $h$-points) o...

$ABCDE$ is a regular pentagon inscribed in a circle, and $A'$ is the other extremity of the diameter...

Two coplanar circles $S$ and $S'$ are exterior to one another and have different radii. A line is ca...

$P, Q, R$ are any points on the sides $BC, CA, AB$ respectively of the triangle $ABC$. Prove that th...

Let $C_1, C_2$ be non-intersecting circles with centres $O_1, O_2$ respectively and common tangents ...

$C$ is the mid-point of $OD$ and the point $Q$ lies on the semi-circle through $D$, with centre $O$,...

Two circles $\Gamma$ and $\gamma$ (lying inside $\Gamma$) of radii $R$ and $r$, respectively, whose ...

From the circumcentre $S$ of a triangle $ABC$, perpendiculars $SD$, $SE$ and $SF$ are drawn to the s...

$C$ is a circle with centre $O$ and radius $R$, $C'$ a circle with centre $O'$ and radius $r$ ($< \f...

$AB$ is the segment $0 \leq x \leq 1$; at each point $P$ of $AB$ whose distance from $A$ is of the f...

A \emph{plane convex set} is a set of points in a plane such that any point of the line-segment join...

In a plane three circles of equal radii are drawn through a point. Prove that the circle through the...

$ABC$ is a triangle, whose angles are $3\alpha, 3\beta, 3\gamma$. Points $P, Q, R$ interior to the t...

If $A, B$ are points in the plane, the part of the line $AB$ between $A$ and $B$ is the segment $AB$...

Let $P_1 P_2 \ldots P_n$ be a regular polygon. Construct points $Q_1$, $Q_2$, $\ldots$, $Q_n$ such t...

Let $P$ be a point on the circumcircle of the triangle $ABC$, and let $L$, $M$ and $N$ be the feet o...

Let $l$ be a fixed line in the plane. Let $P$, $Q$ be distinct points not on $l$ lying on the same s...

A Euclidean motion $M$ of the plane is a transformation of the plane onto itself of the form of a ro...

A set of points in the plane is $k$-distant if the distances $d(A_i, A_j)$ ($i \neq j$) take precise...

The churches of St Aldate, St Buryan and St Cett stand on the flat East Anglian plane, and their tal...

Let $P$ and $Q$ be points on the same side of a line $l$. Let $Q'$ be the reflection of $Q$ in $l$. ...

Two triangles in a plane ($ABC$, $A'B'C'$) are in perspective from a point $O$ (i.e. $AA'$, $BB'$, $...

In each of the following cases either prove the statement true, or give a counter-example to show it...

Imagine that you are provided with a straight-edge and a parallel ruler (which is a device by means ...

$ABC$ is a triangle, $O$ a point inside it. Prove that $$\lambda(BC + CA + AB) > OA + OB + OC > \mu(...

$C_1$ and $C_2$ are two circles; the polars of a point $A$ with respect to $C_1$ and $C_2$ meet at $...

A point $P$ is given and two lines $l$, $m$ whose point of intersection $Q$ is off the paper. You ar...

$AB$, $AC$ are two equal line segments, meeting at an acute angle. $X$ is a point such that $AB$, $A...

Given a triangle $ABC$, points $Q$, $M$ are taken on the side $AC$ such that $AQ = \frac{1}{4}AC$, $...

Two points $A$, $B$ lie on a given circle; $C$ is a point on one arc $AB$ and $D$ is a point on the ...

A triangle $ABC$ suffers two displacements in its plane: (i) a reflexion about a point $O$ to a posi...

Points $P$, $Q$, $R$ are taken on the sides $BC$, $CA$, $AB$ respectively of a triangle $ABC$. Prove...

In a tetrahedron $ABCD$, the points $P$, $Q$, $R$ are the feet of the perpendiculars to $BC$, $CA$, ...

Prove the theorem of Pappus that, if $ABC$ and $PQR$ are two straight lines, then the points of inte...

An `algebra of coplanar points' is constructed as follows: $A$, $B$, $C$, $\ldots$ are points in a p...

An acute-angled triangle $ABC$ is inscribed in a circle; another circle through $B$ and $C$ meets $A...

Three points $A$, $B$, $C$ form an acute-angled triangle in space. Establish the existence of two po...

Two circles intersect in distinct points $A$, $B$; a variable chord through $A$ meets one circle aga...

Two perpendicular straight lines meet at $O$; a circle of centre $P$ cuts the first line in points $...

$P$ is a point in the plane of a triangle $ABC$, not lying on any side of the triangle. The point $P...

The diagonals $AC$, $BD$ of the cyclic quadrilateral $ABCD$ meet in $O$, and $L$, $M$ are the feet o...

$O$ is a point in the plane of a circle $C$, lying outside $C$. $P$ is a variable point on $C$, and ...

$ABC$ is a triangle, with vertices ordered in a counter-clockwise sense. Show that the resultant of ...

$A$, $B$, $C$, $D$ are the points $(r\cos\theta, r\sin\theta)$, for $\theta = \alpha, \beta, \gamma,...

In a triangle $ABC$, $G$ is the centroid and $A'$ is the mid-point of $BC$. The circles $CA'G$, $BA'...

$ABC$ is a triangle. Points $D$, $E$, $F$ are chosen on $BC$, $CA$, $AB$ such that $AD$, $BE$, $CF$ ...

$ABCD$ is a trapezium, with $AB$ parallel to $DC$. Lines $BL$, $DM$ are drawn to $AC$, meeting the r...

$P$ and $Q$ are two points on a semicircle whose diameter is $AB$; $AP$ and $BQ$ meet in $N$. Prove ...

The vertices $P$, $Q$, $R$ of a triangle $PQR$ lie on the sides $BC$, $CA$, $AB$ respectively of a f...

$A$, $B$, $C$, $D$ are four points on a circle $S$. $BC$ and $AD$ meet in $X$, $CA$ and $BD$ meet in...

$ABC$, $A'B'C'$ are two skew lines, and $AB:BC = A'B':B'C'$. Prove that the mid-points of $AA'$, $BB...

Prove that the circumcircles of the four triangles formed by the sets of four lines in general posit...

$ABC$ is an acute-angled triangle and $BC$ is its shortest side. The altitude from $A$ to $BC$ is of...

$ABC$ is a triangle, $S$ its inscribed circle, and $S_1$, $S_2$, $S_3$ the three escribed circles. S...

Circles are drawn through a fixed point $A$ to cut a fixed line $l$, not passing through $A$, at a f...

$X$, $S$ are opposite ends of the diameter of a circle $C$ and $l$ is the line tangent to $C$ at $N$...

$S$ is the inscribed circle of a triangle $A_1 A_2 A_3$, and $S_1$, $S_2$, $S_3$ are the three escri...

A line segment $AB$ of constant length $b$ is such that $A$ lies on the line $y = 0$ while $AB$ (pro...

The diagonals $A_1 A_3$, $A_2 A_4$ of a quadrangle $A_1 A_2 A_3 A_4$ intersect at right angles at $O...

Two circles intersect in $A$ and $B$. [A convenient figure is obtained by taking the radii to be app...

The altitudes $AP$, $BQ$, $CR$ of an acute-angled triangle $ABC$ meet in the orthocentre $H$, where ...

A point $P$ lies in the plane of a given triangle $XYZ$. The lines $XP$, $YP$, $ZP$ meet $YZ$, $ZX$,...

Prove that if $A$, $B$, and $C$ are three collinear points and $P$ is a point not on the same straig...

Two equal circles touch each other externally at a point $O$, and the tangent at a general point $P$...

If for a triangle $ABC$ the circumcentre is $O$ and the orthocentre is $H$, show that $$OH^2 = R^2(1...

The sides $AB$, $BC$, $CD$, $DA$ of a plane quadrilateral are of lengths $a$, $b$, $c$, $d$, respect...

If $P$ is a point on the circumcircle of a triangle $ABC$ and $L$, $M$, $N$ are the feet of perpendi...

$\Delta_n (-\infty < n < \infty)$ is a sequence of triangles, the vertices of $\Delta_{n+1}$ being t...

A quadrilateral has sides $ABC$, $AB'C'$, $A'BC'$ and diagonal lines $A'B'C'$, $A'B'C$ and $XYM$. By...

$ABC$ is a triangle and $O$ any point, not necessarily in its plane. The points $L$, $M$, $N$ divide...

If the lengths of the sides of a quadrilateral are given, show that the quadrilateral has maximum ar...

Prove that the area of the greatest equilateral triangle which can be drawn with its three sides pas...

A navigator wishes to determine the position $D$ of his ship; he observes three landmarks $A$, $B$, ...

Assuming that the length of the circumference of a circle lies between the total lengths of side of ...

$X$, $Y$ are fixed points of a circle and the tangent at a variable point $A$ of the circle meets th...

$R$ is the radius of the circumcircle of the triangle $ABC$. Show that the distance between the orth...

Prove that the quadrilateral of greatest area with sides of prescribed lengths is cyclic. A closed c...

If $\Gamma$ is a circle with centre $C$, and $A, B$ are two points in the same plane as $\Gamma$ (bu...

A triangle is to be circumscribed around a given circle. Prove that, if it is to have the minimum ar...

All three angles of the triangle $ABC$ are less than $120^\circ$. Show that the minimal value of $PA...

Show that the product of two involutions is another involution if and only if the double points of t...

Let $A$, $B$, $C$, $D$ be four given points in a plane, no three of them being collinear. Suppose th...

Prove Desargues' theorem that, if the lines joining corresponding vertices of two coplanar triangles...

The inscribed circle $\Gamma$ of a triangle $ABC$ touches the sides of the triangle at $D$, $E$, $F$...

On a level plain are to be seen three church steeples of different heights. Three men walk on the pl...

$ABC$ is an acute-angled scalene triangle, whose incentre is $I$ and circumcentre is $O$. Prove that...

Let $O$, $U$, $A$, $B$ be distinct points on a line $l$; $a$, $b$, $u$ lines through $A$, $B$, $U$ i...

Prove that the arithmetic mean of $n$ positive numbers is not less than their geometric mean. Prove ...

Explain what is meant by an involution of pairs of points on a line. A line $p$ meets the sides $BC$...

$a, b, c, d$ and $l$ are five coplanar lines, no three of which are concurrent, and $E, F, G$ are th...

If $l,m,p$ and $q$ are real numbers and $lm<0$, show that the equations \[ xy=p, \quad (y-lx)(y-mx)=...

Prove Desargues' theorem, that if two triangles in the same plane are in perspective from a point th...

If three straight lines do not all lie in one plane, prove that, in general, there are infinitely ma...

Three points $A, B, C$ are given on a line $l$. A fourth point $D_1$ of the line is determined by th...

Points $D, E, F$ are given on the respective sides $BC, CA, AB$ of a triangle $ABC$ such that \[ \fr...

Four points $A,B,C,D$ lie on a circle. The orthocentres of the triangles $BCD, ACD, ABD, ABC$ are $P...

A general point $O$ is taken in the plane of a triangle $ABC$; the lines $AO, BO, CO$ meet $BC, CA, ...

P, Q, R are three collinear points, and O is a point not on the line PQR. Lines are drawn through P,...

Through the vertices $A, B, C$ of an acute-angled triangle $ABC$ straight lines $VAW, WBU, UCV$ are ...

$ABCD$ is a plane quadrilateral. The line through $A$ parallel to $BC$ meets $BD$ in $P$, and the li...

A quadrilateral $ABCD$ varies in such a manner that it is always inscribed in a fixed circle, of cen...

A point $D$ is taken on the minor arc $BC$ of the circumcircle of an equilateral triangle $ABC$, and...

Given a triangle $ABC$ and a point $P$ on its circumcircle, it is known that the feet of the perpend...

Squares $BCLP, CAMQ, ABNR$, of centres $X, Y, Z$, are described outwards on the sides $BC, CA, AB$ o...

A point $U$ is taken on the circumcircle of a triangle $ABC$, and $P, Q, R$ are the feet of the perp...

Two lines $l, p$ meet in a point $U$. Points $L, M, N$ are taken on $l$ and points $P, Q, R$ are tak...

Given a circle of centre $A$ and a point $O$ outside it, obtain a construction, by ungraduated ruler...

The altitudes $AD, BE, CF$ of an acute-angled triangle $ABC$ meet in the orthocentre $H$, and $O$ is...

$ABC$ is an acute-angled triangle in which $AB > AC$. The internal bisector of the angle $A$ meets $...

A point $P$ is taken on the diagonal $BD$ (for convenience, produced beyond $D$) of the parallelogra...

The point $I$ is the incentre of the triangle $ABC$. Determine under what conditions the bisector of...

Two points $A, B$ are given, and a circle is drawn such that the length of the tangent from $A$ to i...

Lines $\alpha, \beta, \gamma$ are drawn through the respective vertices $A, B, C$ of a triangle $ABC...

Perpendiculars $PX, PY, PZ$ are drawn from an arbitrary point $P$ in the plane to the sides of the t...

From a variable point on a diagonal $WY$ of a parallelogram $WXYZ$ lines are drawn through fixed poi...

A point $P$ moves in a plane so that the ratio of its distances from two fixed points $A$ and $B$ in...

Prove Desargues' theorem that if two triangles in a plane are in perspective the intersections of th...

H is the orthocentre and O the circumcentre of a triangle $ABC$. $AO$ meets the circumcircle again i...

Prove that there is a point $P$ in the plane of a triangle $ABC$ such that the angles $\angle BCP, \...

Prove that the lines joining the mid-points of the three pairs of opposite sides of a quadrangle are...

Two lines in a plane meet in $K$. Prove that successive reflection in the two lines is equivalent to...

Prove that a point lies on the circumcircle of a triangle if and only if the feet of the perpendicul...

The equations of the sides of a triangle referred to rectangular Cartesian axes are \[ u_i = a_ix+b_...

A straight line meets the sides $BC, CA, AB$ of a triangle at $L, M, N$ respectively. Prove that the...

The lines $a,b,c$ in a plane are concurrent at $V$; the pairs of points $A$ and $A'$, $B$ and $B'$, ...

Points $X, Y, Z$ lie respectively on the sides $BC, CA, AB$ of the triangle $ABC$ in such a way that...

Prove that the Simson line of a point D on the circumcircle of a triangle ABC bisects the join of D ...

Prove that if in the tetrahedron ABCD, AB=CD and AD=BC, then AC and BD are bisected by their mutual ...

The two diagonals AC and BD of a plane quadrilateral meet in O. Prove that \[ \text{area } \triangle...

The lines joining a point $O$ in the plane of a triangle $ABC$ to the vertices meet the sides $BC, C...

If the triangle $ABC$ has sides of length $a,b$, and $c$, respectively, and if with the usual notati...

$ABCD$ is a square of side $a$. A point $P$ moves so that the sum of the squares of its distances fr...

Given one vertex $A$, the circumcentre $O$, and the orthocentre $H$ of a triangle, show how to const...

Prove the Simson's Line theorem for a triangle inscribed in a circle, namely that the feet of perpen...

Two points $X, Y$ of general position are taken in the plane of a fixed circle $C$. Obtain a constru...

The circumcircle of an obtuse angled triangle $ABC$ subtends an angle $2\theta$ at the orthocentre. ...

$A, B, C,$ and $D$ are four generally placed coplanar points. $AD$ and $BC$ meet in $X$, $AC$ and $B...

Given the values $r_a, r_b,$ and $r_c$ of the radii of the escribed circles of a triangle, find in t...

The vertex $A$ of a triangle is at a fixed point of a given circle with centre $O$. The base $BC$ is...

A square $PQRS$ of side $x$ is inscribed in a triangle $ABC$ in such a way that $PQ$ lies on the sid...

If two triangles are in perspective from a point, prove that the three points of intersection of pai...

Prove that, if the joins of corresponding vertices of two coplanar triangles are concurrent, the int...

(i) Two coplanar triangles $PQR$ and $P'Q'R'$ are in perspective. $L$ is the point of intersection o...

The altitudes of an obtuse-angled triangle $ABC$ intersect at a point $H$. Prove that the circumcirc...

A leaf of a book is of width $a$ in. and height $b$ in., where $3a \le 2\sqrt{2}b$; the lower corner...

Prove that the area of the triangle whose sides are $a, b, c$ is $\sqrt{s(s-a)(s-b)(s-c)}$, where $2...

$P$ is a point inside a triangle $ABC$, at distances $a', b', c'$ from $A, B, C$ respectively; the a...

$ABCD$ is a cyclic quadrilateral whose diagonals $AC, BD$ meet in $X$. $E$ and $F$ are the feet of t...

The mid-points of the sides $AB, CD$ of a parallelogram $ABCD$ are $X, Y$. $P$ is a point on the dia...

The triangle $ABC$ lies entirely inside the triangle $DEF$. Show that the sum of the sides of $ABC$ ...

A region $\mathcal{R}$ of the plane is defined to be \textit{convex} if for each pair of points $A, ...

A polygon $P$ has vertices $A_1, \dots, A_n$ where the coordinates $x_r, y_r$ of $A_r$ are both inte...

The triangle $ABC$ is acute-angled; $P$ is a point that can vary on $BC$ (but not outside the segmen...

Three coplanar circles $\alpha, \beta, \gamma$ have a common point $O$. The common chord $PO$ of $\b...

Prove that, if two pairs of opposite edges of a tetrahedron are at right angles, so is the third pai...

$P, A, B, C$ are four points in space. Through the mid-points of $BC, CA, AB$, lines are drawn paral...

Lines drawn from the vertices $A, B, C$ of a triangle through a variable point $O$ within the triang...

A variable circle through two fixed points $A$ and $B$ cuts a fixed circle at $P$ and $Q$. Prove tha...

Points $X, Y, Z$ are taken on the sides $BC, CA, AB$ of a triangle. Prove that $AX, BY, CZ$ are conc...

D, E, F are the middle points of the sides BC, CA, AB respectively of the triangle ABC, X is any poi...

State (without proof) a construction for (i) the radical axis, (ii) the limiting points of a coaxal ...

$P$ is a point on the circumcircle of the triangle $ABC$, and the lines through $P$ perpendicular to...

Prove that the common chords of three (intersecting) circles taken in pairs are concurrent. $D, E, F...

Prove that the Simson's line of a point $P$ on the circumcircle of a triangle $ABC$, with respect to...

From a point $O$ on the circumcircle of a triangle $ABC$, lines $OL, OM, ON$ are drawn perpendicular...

Two given circles cut orthogonally at $A$ and $B$. A third circle is drawn through $A$ to cut them i...

D, E, F are the mid-points of the sides BC, CA, AB of a triangle, Y and Z are the feet of the perpen...

Three concurrent lines OA, OB, OC are cut by a transversal ABC. P and Q are two points on OA; PB mee...

Prove that the circumcircles of the triangles formed by sets of three out of four given lines meet i...

A straight line $l$ meets the sides $BC, CA, AB$ of a triangle in $A_1, B_1, C_1$ respectively. $O$ ...

A circle $S$ is described on $AB$ as diameter, and $CD$ is any chord of $S$. The line through $A$ pe...

Four points $A, B, C, D$ are coplanar. $AD$ and $BC$ meet in $P$, $BD$ and $CA$ meet in $Q$, and $CD...

Prove Pappus' Theorem that, if $A_1, B_1, C_1$ and $A_2, B_2, C_2$ are two sets of three collinear p...

If $ABC$ is an acute-angled triangle, show how to construct the point $P$ at which all the sides sub...

$A_1, A_2, A_3, A_4$ are the vertices of a quadrangle; $G_1$ is the centroid of $A_2A_3A_4$; $G_2, G...

$A_1, A_2, B_1, B_2$ are four points in space. $C_1$ divides $A_1B_1$ in the ratio $\lambda:1$ and $...

$ABC$ is a triangle; $PQR$ is inscribed in $ABC$, $P$ lying on $BC$, $Q$ on $CA$ and $R$ on $AB$. Pr...

P is a point in the plane of the triangle ABC, and L, M, and N are the feet of perpendiculars from P...

Pairs of points $(P_r, Q_r)$ on a given straight line $l$ are chosen so that $AP_r, BQ_r$ intersect ...

A', B', C' are any points on the sides BC, CA, AB respectively of triangle ABC. Prove that...

Four unequal similar triangles can be drawn with sides touching a given circle of radius $\rho$. ...

Prove that the feet of perpendiculars from a point $P$ of the circumcircle of the triangle $ABC$ on ...

Prove that for a tetrahedron: \begin{enumerate} \item[(i)] The joins of the midpoints of opposite ...

Establish the existence of the Nine-Point circle of a triangle and prove Feuerbach's Theorem that th...

Prove that the area $A$ of a convex plane quadrilateral whose sides are of length $a,b,c,d$ is given...

Prove that the locus of a point moving with its distances from two fixed points in a constant ratio ...

In a triangle $ABC$ the inscribed circle touches the sides $BC, CA, AB$ at $A_0, B_0, C_0$ respectiv...

If $I$ is the incentre of the triangle $ABC$, prove that $AI$ passes through the circumcentre of the...

Prove Apollonius' theorem, that if $D$ is the mid-point of the base $BC$ of a triangle $ABC$, then $...

$X$ is the point inside a triangle $ABC$ such that $XB, XC$ are the internal trisectors of the angle...

Establish the existence of the Nine Points Circle of a triangle $ABC$, and determine the position of...

$\alpha, \beta, \gamma$ and $\alpha', \beta', \gamma'$ are the sides of two triangles circumscribed ...

Define the radius of curvature at a general point of a plane curve, and from the definition derive t...

A point $P$ is selected in the plane of a fixed triangle $ABC$ and a function of the position of $P$...

Prove that, if an angle of a triangle and the length of the opposite side and the length of the bise...

Shew that a triangular prism with parallel plane ends can be divided into three tetrahedra of equal ...

Upon a given line as base and upon the same side of it six triangles may be constructed equiangular ...

Two triangles $\Delta$ and $\Delta'$ are inscribed in the same circle, and in each a vertex and the ...

The side of a hill is an inclined plane with slope of 1 in 30. A level railway running along the sur...

A rectangular tank 6~ft. long, 5~ft. wide and 4~ft. deep stands on a slope with the two corners at t...

The opposite sides of a quadrilateral inscribed in a circle meet in $P$ and $Q$. Prove that the bise...

A plane cuts off from a sphere a volume equal to $\frac{7}{27}$ of the whole. Find the ratio in whic...

A triangle $ABC$ is inscribed in a circle, and chords $Aa, Bb$ are drawn parallel to the sides $BC, ...

The angles of a parallelogram are bisected externally: prove that the bisectors form a rectangle who...

$ABC$ is a triangle; $D, E, F$ are the feet of the perpendiculars from $A, B, C$ on the opposite sid...

Prove that from a point $h$ feet above the surface of the sea the distance to the horizon is $1\cdot...

The sides $BC$, $CA$, $AB$ of a triangle $ABC$ are $3x+2y=39$, $2x-y=5$, $9x-y=33$, respectively. Sh...

$P$ is a point inside a quadrilateral $ABCD$ such that the sum of the areas $PAB, PCD$ is constant. ...

$A$ and $B$ are the centres of two circles which intersect in $P$ and $Q$; the angle $APB$ is less t...

$ABC$ is a triangle obtuse-angled at $A$; $D$ is the foot of the perpendicular from $A$ on the side ...

$AB$ is a fixed chord of a circle, and $KL$ is a variable chord of fixed length; $AK$ and $BL$ inter...

A cyclic quadrilateral $ABCD$ is such that a circle can also be inscribed in it. If the sides $AB, B...

$AB, BC$ are adjacent sides of a regular polygon, $O, D$ are the middle points of $AB, BC$, respecti...

$ABC$ is a triangle, $D$ the middle point of $BC$; $DG$ is drawn to cut the circle $ABC$ in $G$ and ...

$A, B, C$ are fixed points. It is required to find a point $P$ in the plane $ABC$ such that $PA:PB:P...

If the medians from $B$ and $C$ of a triangle $ABC$ are inclined at an angle $\frac{1}{3}\pi$, then ...

$ABC$ is a triangle, $P$ any point on the circumscribing circle. Shew that the feet of the perpendic...

$ABC$ is a triangle. Shew that the increases in area resulting from small increases $\delta a, \delt...

If two circles cut at right angles shew that the intercept made by either circle on any line drawn t...

Given a straight line $AB$ divided into two segments by a point $P$ shew that the locus of points at...

The internal and external bisectors of the angle $A$ of the triangle $ABC$ are drawn meeting $BC$ in...

Two circles intersect in $A, B$. Through $A$ a straight line $CAD$ is drawn cutting the circles in $...

$ABC$ is a triangle, $P$ any point on the internal bisector of the angle $BAC$; $BP, CP$ are produce...

A triangle with sides 5, 5, 6 has three circles inscribed in it each touching the other circles and ...

$ABC$ is an equilateral triangle inscribed in a circle of radius $a$; $P$ is any point on a concentr...

If $O$ is a point inside a triangle $ABC$, and $A'$, $B'$, $C'$ are the feet of the perpendiculars f...

A side $a$ and the opposite angle $A$ of a triangle $ABC$ are measured and found to be 6 inches and ...

The inscribed circle of the pedal triangle $DEF$ of a triangle $ABC$ touches the sides $EF, FD, DE$ ...

A sphere rolls on a parabolic wire with which it is in contact at two points; shew that the locus of...

Points $P$ and $Q$ are taken upon two opposite sides $AB$, $CD$ of a square $ABCD$. Shew that, if th...

Through the intersection of the diagonals of a quadrilateral lines are drawn parallel to the four si...

A segment is cut of the parabola $y^2=4ax$ by a chord joining the points $(x_1, y_1)$ and $(x_2, y_2...

Find the approximate increment in the radius of the circumscribed circle of a triangle $ABC$ when th...

The reflexions of the vertices $A, B, C$ of a triangle in the opposite sides are $A'$, $B'$, $C'$. A...

A straight line meets the sides $BC, CA, AB$ of a triangle in $L, M, N$. The parallelograms $MANP, N...

Each edge of a tetrahedron $OPQR$ is equal to the opposite edge, and $A, B, C$ are inverse to $P, Q,...

A variable straight line through the centre $O$ of a regular hexagon $ABCDEF$ meets $AC$ in $G$ and ...

$A_1, A_2, A_3, A_4$ are four coplanar points such that the line joining any two is perpendicular to...

$l_ix + m_iy + n_i = 0$, ($i=1,2,3$), are the equations of three lines. $N_i$ is the cofactor of $n_...

Four straight lines are given; prove that a system of three circles can be found in an infinite numb...

A rectangular sheet of paper $OACB$ is folded over so that the corner $O$ just reaches a point $P$ o...

Shew that the four circles which circumscribe the triangles formed by three out of four given lines ...

State and prove Menelaus' Theorem. Prove that the centres of similitude of three circles (in the sam...

A quadrilateral whose sides are of lengths $a,b,c,d$ is inscribed in a circle. Prove that the length...

Prove that the circumcircles of the four triangles formed by four coplanar lines meet in a point $O$...

Prove that, if the lines joining corresponding vertices of two triangles $ABC, A'B'C'$ are concurren...

Prove that the joins of mid-points of opposite edges of a tetrahedron meet in a point. Shew that, if...

Prove that if $ABCD$ is a quadrilateral then in general the sum of the rectangles $AB.CD$ and $BC.AD...

Prove that, if the perpendiculars from $A'$, $B'$, $C'$ to the sides $BC$, $CA$, $AB$ of the triangl...

Two curves $C_1$, $C_2$ and a point $P$ common to them are inverted with respect to any circle whose...

$OABC, OA'B'C'$ are two straight lines; $AB', BA'$ meet at $P$; $BC', CB'$ meet at $Q$, and $CA', AC...

$A$ is a fixed point on a sphere and $P$ is a variable point on it. $AP$ is produced to $Q$ so that ...

Prove that the inverse of a circle is either a circle or a straight line. Two fixed circles $C$ ...

Two pencils, with vertices $A$ and $B$, are homographically related in such a way that the ray $AB$ ...

$X$ and $Y$ are any points of the line $AB$, and $X'$, $Y'$ are their harmonic conjugates with respe...

Show that the locus of a point $P$ in space whose distances from three fixed points $A, B, C$ are in...

The points $D, E, F$ lie on the sides $BC, CA, AB$ respectively of a triangle $ABC$. Prove that a ne...

Two fixed points $A, B$ lie on a given tangent to a conic $S$. $P$ is the pole with regard to $S$ of...

Through a given point inside a parallelogram construct a straight line which shall divide the area o...

A, B, C, D, E are five points in space, no four lying in the same plane. From each of the five point...

Spheres are described to touch two fixed planes and to pass through a fixed point. Prove that they a...

A square $PQRS$ lies in a given plane, and the sides $PQ, QR, RS$ (produced if necessary) pass, resp...

$ABC$ is a given triangle and $P$ is a general point in its plane. The lines $PA, PB, PC$ meet $BC, ...

Three collinear points $A, B, C$ are given. Give a construction, using a straight edge only, for the...

A tetrahedron $ABCD$ is such that $AB=CD$, $AC=BD$, $AD=BC$. Prove that (i) the lengths of the perpe...

A point $O$ moves on the line which bisects the angle $C$ of a triangle $ABC$, and $AO, BO$ produced...

Two ships are steaming along straight courses which converge at an angle of $60^\circ$. If their dis...

$ABC$ is an acute-angled triangle, $D, E, F$ are the middle points of the sides $BC, CA, AB$ respect...

Shew that a uniform flexible chain hangs under gravity in a catenary whose Cartesian equation can be...

From a point $P$ outside a circle two lines $PAB, PDC$ are drawn, cutting the circle at $A, B, C, D$...

From the angular points $A, B, C$ of an equilateral triangle, whose side is 3 inches, lines $AP, BQ,...

A plane is drawn dividing a sphere into two parts whose volumes are in the ratio $3:1$. If $2\alpha$...

$ABC$ is a triangle inscribed in a circle, the tangents at $B$ and $C$ meet at $T$. Shew that, if a ...

A point $A$ moves along a straight line $a$ and is joined to two fixed points $B$ and $C$ such that ...

The planes of two intersecting circles of radii $a$ and $b$ are inclined at an angle $\alpha$, and t...

In a triangle $ABC$ the side $AB$ and the distance from $C$ to the middle point of $AB$ are accurate...

If the sides of a parallelogram are parallel to the lines $ax^2+2hxy+by^2=0$ and one diagonal is par...

Shew that the feet of the perpendiculars drawn from a point on the circumscribing circle to the thre...

Prove that the sum of the squares of the medians of a triangle $ABC$ is $\frac{3}{4}(a^2+b^2+c^2)$. ...

The bisectors of the angles of the triangle $ABC$ cut the opposite sides in $D, E, F$. Find the leng...

State and prove Menelaus' theorem on transversals. In the triangle $ABC$, $AB=AC$ and $DEF$ is a t...

$ABC$ is a triangle and $AD$ the perpendicular on $BC$. Obtain a formula for $\cos A$ in terms of th...

A triangle whose angles are 47$^\circ$, 71$^\circ$, and 62$^\circ$ is inscribed in a circle of radiu...

Prove that the line joining a point $P$ on the circumcircle of a triangle to the orthocentre of the ...

$ABCDEFG$ is a regular heptagon inscribed in a circle of radius 1. Shew that the distance between th...

$ABC$ is any triangle, $X$ any point. Shew that there exists a point $X'$ such that \[ B\hat{A}X' ...

If A, B, C, D are four concyclic points, shew that the feet of the perpendiculars from D on the side...

Prove one of the following theorems and deduce the other from it. \begin{enumerate} \ite...

Explain the general principles of the method of inversion in pure geometry, and state and prove what...

Three light strings are attached at points $A$, $B$, $C$ to a circular hoop which is in a vertical p...

Obtain the equations \[ y = c \cosh \frac{x}{c}, \quad s = c \sinh \frac{x}{c} \] for the fo...

In any triangle prove that \[ r=4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}. \] If a poi...

Prove that the area of the triangle formed by joining the feet of the perpendiculars from the corner...

Shew how to construct a triangle when the centres of the inscribed circle, of the circumcircle and o...

The internal bisectors of the angles $A, B, C$ of a triangle meet the circumcircle in $A', B', C'$. ...

Find the sides of the pedal triangle of a triangle $ABC$ in terms of the sides of $ABC$. $L, M, N$ ...

The internal bisector of the angle $A$ of a triangle $ABC$ meets $BC$ in $D$; prove that \[ AD = \fr...

Prove that the radius of the inscribed circle of a triangle $ABC$ is equal to $a \sin\frac{1}{2}B \s...

In any triangle $ABC$ prove that the sum of the squares of the distances of the centre of the inscri...

The internal bisectors of the angles of the triangle $ABC$ (with sides $a,b,c$ and area $\Delta$) me...

In walking a mile up the line of greatest slope of an inclined plane a man finds that he has risen 3...

With the usual notation for the radii of the inscribed and escribed circles of the triangle $ABC$, p...

(i) If $I$ be the in-centre and $O$ the circumcentre of a triangle $ABC$, shew that \[ OI^2 = R^2 -...

In a triangle $ABC$, it is given that the line joining the orthocentre $H$ and the circumcentre $O$ ...

Prove that, if $H$ and $O$ are the orthocentre and circumcentre of a triangle $ABC$, \[ OH^2=R^2(1-8...

(i) The radii of the escribed circles of a triangle are $r_1, r_2$ and $r_3$, the radius of the insc...

The points $P_1, P_2, \dots, P_n$ are the vertices of a regular $n$-agon inscribed in a circle $C_0$...

The feet of the perpendiculars from a point $P_1$ to the sides of a triangle $ABC$ lie on a straight...

Prove that if chords $AA', BB', CC'$ of a circle are concurrent the products $BC' \cdot CA' \cdot AB...

Prove that there are four plane sections of a cube which are regular hexagons. Shew that a flexi...

The distances of a point from the vertices of an equilateral triangle of unknown size are given. Sho...

Prove that the radical centre of the three escribed circles of a triangle is the centre of the circl...

Prove that the circumcircle of a triangle passes through the focus of any parabola which touches its...

The sides of a plane polygon $A_1A_2A_3\dots A_n$ are cut by a straight line in the points $B_1, B_2...

Shew that any transversal cuts a plane pencil of four fixed lines in a range of constant anharmonic ...

Interpret the equations: \begin{enumerate} \item[(1)] $\lambda S_1 + \mu S_2 = 0$, ...

Prove that, if the middle points of the coplanar lines $AB, BC, CD, DA$ are concyclic, $AC$ is at ri...

Prove that the locus of a point, the lengths of the tangents from which to two fixed circles are in ...

In a plane a circle is given and two points external to it. Shew how to construct the two circles wh...

If $P$ is a point in the plane of the triangle $ABC$ and $\alpha.PA^2 + \beta.PB^2 + \gamma.PC^2 = \...

Shew how to draw a line through a given point to meet two given non-intersecting lines. If $A, B...

Prove that there are in general two points $P$ in the plane of a triangle $ABC$, such that $PA:PB:PC...

Given the circumcentre, the orthocentre and one vertex of a triangle, shew how to determine the othe...

Prove that $OI^2 = R^2 - 2Rr$, where $O, I$ are the centres of the circumscribed and inscribed circl...

$ABC$ is a triangle in which the angles $ABC, ACB$ are each equal to twice the angle $BAC$. Prove th...

Prove that the orthocentre $H$, the centroid $G$ and the centre $O$ of the circumcircle of a triangl...

(i) Find the angle between the straight lines given by the equation (in rectangular Cartesian coordi...

Two unequal circles, lying in different planes, meet in two points, $A$ and $B$. Shew that there is ...

Find equations for the incentre of the triangle formed by the lines \[ x-2y=0, \quad 4x-3y+5=0, \qua...

Two circles, with centres $A$ and $B$ and radii $a$ and $b$, lie in different planes which meet in a...

$P$ is any point in the plane of a triangle $ABC$, and $X$ is the reflexion of $P$ in the side $BC$ ...

P, Q and R are any three points. The circle C on QR as diameter meets PQ in Q' and PR in R'. Show th...

$ABC$ is a triangle whose angle $A$ is a right angle. Lines parallel to the opposite sides are drawn...

If the diagonals of a quadrilateral inscribed in a circle are perpendicular to each other, prove tha...

Prove that there are two real points P, Q in space at each of which the sides of a given acute-angle...

The inscribed circle of a triangle $ABC$ touches the side $BC$ at $X$ and the inscribed circles of t...

If $P$ is a variable point on a fixed circle and $O$ is any point not in the plane of the circle, pr...

The sides of a triangle lie along the lines $u \equiv x\cos\alpha+y\sin\alpha-p=0$, $v \equiv x\...

A variable sphere passing through a fixed point touches each of two fixed spheres; prove that the lo...

Shew that the necessary and sufficient condition that the three pairs of points $A, A'; B, B'; C, C'...

A tetrahedron has each edge perpendicular to the opposite edge. Prove that the four perpendiculars f...

The angles of any triangle $ABC$ are trisected and the two trisectors nearest to the side $BC$ meet ...

In a quadrilateral $ABCD$ the sides are $AB=a, BC=b, CD=c, DA=d$; and the angle $DAB=\theta, ABC=\ph...

Two straight lines are given by the equations \[ p = ax+by+c=0, \quad p' = a'x+b'y+c'=0; \] ...

Explain the geometrical method known as generalization by projection, and generalize the following r...

If $a, b, c, d$ are four coplanar lines, prove that \begin{enumerate} \item the circumcircles ...

If $ABC$ is a triangle self-polar with respect to a conic $S$, and if $\alpha$ is the polar of anoth...

Shew how to construct a mean proportional to two given straight lines and prove the validity of your...

Draw a diagram to illustrate the truth of the algebraical identity \[ (a-b)(a+b) = a^2-b^2. \] ...

Illustrate by a figure the truth of the identity \[ a^2-b^2 = (a-b)(a+b). \] \item[*3.] If a...

Prove that, if the straight lines joining a point $P$ to the vertices of a triangle $ABC$ meet the o...

The vertices $B$ and $C$ of a triangle are fixed and the angle $A$ is given. shew that the vertex $A...

Shew how to construct an isosceles triangle of given size such that each of the angles at the base i...

Shew that the general equation of the first degree in Cartesian coordinates represents a straight li...

Shew that if the opposite edges of a tetrahedron are at right angles then the perpendiculars from th...

The inscribed circle of the triangle $ABC$ touches $BC$ at $D$, $CA$ at $E$ and $AB$ at $F$; $P$ is ...

If the diagonals of a quadrilateral intersect at right angles at $O$, shew that the feet of the perp...

Prove that if corresponding sides of two coplanar triangles meet in three collinear points, their co...

$D, E, F$ are the feet of the perpendiculars from the vertices on the opposite sides of the triangle...

In a triangle prove that, with the usual notation, \begin{enumerate} \item $1/r_1 + 1/r_2 + 1/...

State and prove the property from which the nine points circle of a triangle derives its name. $...

In any triangle, prove that the centre of the nine-points circle bisects the straight line joining t...

Three similar triangles $PBA, AQB, BAR$ are described on the same side of $AB$, the similarity being...

The sides $BC, CA, AB$ of a triangle $ABC$ are cut by a straight line in $D, E, F$ respectively. Pro...

$ABCD$ is a tetrahedron. By drawing pairs of parallel planes through the pairs of opposite edges a p...

Prove the property of the ``Nine Point'' circle of a triangle. Shew that if through the mid-points ...

Explain what is meant by a system of Coaxal Circles. Shew that any straight line is cut by the circl...

For a triangle $ABC$, $R$ is the radius of the circumscribed circle, and $r_1$ the radius of the esc...

Prove that the radius $R$ of the circle that touches externally each of three circles of radii $a, b...

$APQB$ is a straight line, and the lengths of $AQ, PB$ and $AB$ are $2a, 2b$ and $2c$ respectively. ...

A convex quadrilateral of sides $a,b,c,d$ is inscribed in a circle of radius $R$. Prove that \[ ...

Shew that, if the equation of a circle in areal coordinates is in the form \[ \phi(x,y,z) \equiv...

Prove that with the usual notation \[ Rr = \frac{abc}{4s} \quad \text{and} \quad r_1+r_2+r_3=r+4...

On the diameter of the circumscribed circle which passes through the orthocentre of the triangle $AB...

A diagonal of a quadrilateral makes angles $\alpha, \beta$ with the sides at one of its ends, and an...

From a point on the radius $OA$ of the circumcircle of a triangle $ABC$ perpendiculars are drawn to ...

The radii of two parallel plane sections of a sphere are $a,b$, and the distance between them is $c$...

In a triangle prove that \begin{enumerate} \item[(i)] $r = 4R \sin\frac{1}{2}A \sin\frac...

Find expressions for the sides and angles of the pedal triangle of a triangle ABC. Shew that, if O i...

Given the circumcentre, the nine-point circle and the difference of two angles of a triangle, constr...

If $r, R$ denote the radii of the inscribed and circumscribed circles of triangle $ABC$, the centres...

$P$ is any point on the circumcircle of a triangle $ABC$. $PL, PM, PN$ are drawn perpendicular to th...

Four equal spheres of radius $r$ all touch one another. Find the radius of the smallest sphere that ...

$A'$ is a variable point on the circumcircle of a given triangle $APQ$ such that $A$ and $A'$ lie on...

$D, E, F$ are respectively the feet of the perpendiculars drawn to the sides $BC, CA, AB$ of a trian...

State and prove the harmonic property of the quadrangle. How many points are equidistant from four p...

$A, B, C$ are the vertices of a triangle. If points $C', B'$ are taken in the sides $AB, AC$ respect...

State the theorems of Ceva and Menelaus and prove one of them together with its converse. Co...

Any point $P$ is taken in the plane of a triangle $ABC$. Through the mid-points of $BC, CA, AB$ line...

Three points $L, A, B$ are taken on a circle $S$, and $O$ is the mid-point of $AB$. Prove that the t...

The point $K$ is the other end of the diameter through $A$ of the circumcircle of the triangle $ABC$...

Prove that the inverse of a straight line is a circle through the centre of inversion. Circles $...

Points $D, E, F$ are taken in the sides $BC, CA, AB$ respectively of a triangle $ABC$. Prove that th...

$ABC$ is a triangle, and $X$ a point inside the triangle such that \[ \angle XBC = \tfrac{1}{3}\...

Two isosceles triangles have the same inscribed circle and the same circumscribed circle: prove that...

On the sides of a triangle $ABC$ equilateral triangles $BPC, CQA,$ and $ARB$ are described externall...

Given an obtuse-angled triangle, determine a circle of which it is the self-conjugate triangle. Show...

Prove that in any triangle $ABC$, \[ \cos A + \cos B + \cos C \le \frac{3}{2}, \] \[ \cot B \cot...

A quadrilateral is such that one circle can be described about it and another can be inscribed in it...

Prove that: \begin{enumerate} \item[(i)] $\cos^{-1}\frac{4}{5} = 2\tan^{-1}\frac{1}{3}$;...

A, P, Q, B are four points in order on a straight line. $AQ=2a, PB=2b$ and $AB=2c$. Circles are desc...

In any triangle $ABC$, with the usual notation, prove that \[ r = a \sec\frac{A}{2} \sin\frac{B}{2...

$A_1A_2A_3A_4A_5A_6A_7$ is a regular heptagon, and the lines $A_2A_5, A_2A_6, A_2A_7$ meet $A_1A_4$ ...

Shew that a straight tube whose cross-section is a regular hexagon can be completely blocked by a so...

Two conics inscribed in a triangle $ABC$ touch $BC$ at the same point $P$, touch $CA$ at $Q, Q'$ and...

Give a geometrical construction for finding two lengths, having given their sum and the mean proport...

On opposite sides of a base $BC$ are described two triangles $ABC, BCD$, such that $\angle ABC=30^\c...

$OC$ touches a circle at $C$ and $OAB$ is a chord. Prove that \[ AB:OC :: BC^2-AC^2 : BC.AC. \] ...

$A, B, C$ are three points on a circle, and a line through the pole of $BC$ meets $AB, AC$ in $P$ an...

Shew that, in addition to the nine-point circle of a triangle, there are four circles which touch th...

Prove that the cone joining any point to a circular section of a sphere cuts the sphere again in a c...

Prove that, if the incircle of a triangle passes through the circumcentre, then \[ \cos A + \cos...

A circle cuts the sides of a triangle in $P$ and $P'$, $Q$ and $Q'$, $R$ and $R'$ respectively. Prov...

Prove the harmonic properties of a complete quadrilateral. If $A, P, B$ are three points in a st...

Show that if the sum of the squares on a pair of opposite edges of a tetrahedron is equal to the sum...

Circles PAQ and PBQ intersect in P and Q and the tangents at A and B are parallel. PA intersects the...

Define the "nine-points" circle of a triangle and prove the property from which it derives its name....

If the lines joining corresponding vertices of two triangles are concurrent prove that the points of...

Through a given point $O$ draw three straight lines $OA, OB, OC$ of given lengths so that $A,B,C$ ma...

Describe a circle to pass through two given points and touch (i) a given straight line, (ii) a given...

Given a self-conjugate triangle with respect (i) to a circle, construct the circle; (ii) to a rectan...

A circle is inscribed in a right-angled triangle and another is escribed to one of the sides contain...

Prove the harmonic properties of a complete quadrilateral. $ABCD$ is a quadrilateral, $AB$ and $CD...

(i) If $O$ is the circumcentre of the triangle $ABC$, and if $AO$ meets $BC$ in $D$, prove that ...

If $D,E,F$ are the feet of the perpendiculars from the vertices $A,B,C$ of a triangle $ABC$ on the o...

$ABC$ is an acute angled triangle, $D,E,F$ are the middle points of the sides $BC, CA, AB$ respectiv...

(a) Give a geometrical construction for a circle through two given points which intercepts a given l...

$P, Q, R$ are any points on the sides $BC, CA, AB$ respectively of the triangle $ABC$. Prove that th...

Obtain the equation of the circumcircle of the triangle formed by the three lines \[ ax+by+c=0, \q...

Prove the expressions for the area of a triangle (i) $abc/4R$, (ii) $r^2 \cot\frac{1}{2}A \cot\f...

Prove that the feet of the perpendiculars drawn on the sides of a triangle from any point of the cir...

Find the trilinear equation of the circle which circumscribes the fundamental triangle $ABC$. Pr...

Prove that the locus of the centre of a conic which touches four given straight lines is itself a st...

Define the curvature at a point of a curve and obtain its value when the equation of the curve is gi...

In a triangle $ABC$, with the usual notation, prove that \[ r=4R\sin\frac{A}{2}\sin\frac{B}{2}\s...

Prove that the rectangle contained by the perpendiculars drawn from any point $P$ on a circle to any...

Find the angle between the two straight lines \[ ax^2+2hxy+by^2=0. \] The distance between t...

Prove that the locus of the centre of a circle that bisects the circumferences of two given circles ...

Prove the formulae (i) $4AR=abc$, (ii) $16Q^2R^2 = (\alpha\beta+\gamma\delta)(\alpha\gamma+\delta\...

The centres of the circumcircle and the inscribed circle of a triangle are $O$ and $I$, the radii ar...

If, in any polyhedron, the numbers of solid angles, faces, and edges are respectively $x,y,z$, shew ...

Given three collinear points $A,B,C$, prove that the harmonic conjugate of $B$ with respect to $A$ a...

If $H$ is the orthocentre of a triangle $ABC$ and if $AH$ cuts $BC$ in $D$ and the circumcircle agai...

The base $BC$ of a triangle is given. Find the locus of the vertex $A$ when (i) the sum of the base ...

If $O, H, I, K$ are respectively the centres of the circum-, ortho-, in-, and nine-point-circles of ...

Prove the existence of the nine-point circle for any triangle. \par Shew that the sum of the squ...

If $I$ is the incentre of a triangle $ABC$, prove that the circumcentre of the triangle $BIC$ is col...

A quadrilateral is inscribed in one circle and circumscribed about another circle. Prove that the in...

$A_1A_2\dots A_n$ is a regular polygon of $n$ sides inscribed in a circle of radius $a$. Prove that ...

Prove that the square of the distance between the centres of the inscribed circle and the circumscri...

In a triangle $ABC$ prove that if $P$ is the orthocentre and $O$ the circumcentre \[ PO^2 = R^2(1-8\...

In any triangle prove that \[ r=4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}, \quad s=4R\cos\...

If $\alpha, \beta, \gamma$ be the distances of the centre of the nine-point circle from the vertices...

Show that angles in the same segment of a circle are equal. A rod $PQ$ slides with its ends $P, Q$ o...

Prove that, if a circle cuts two circles orthogonally, its centre lies on their radical axis. Pr...

Find the angle between the straight lines whose equation is \[ ax^2+2hxy+by^2=0. \] Prove th...

$P, Q, R$ are points on a rectangular hyperbola. Prove that the centre of the hyperbola lies on the ...

Prove that the feet of the perpendiculars let fall from a point on the circumcircle of a triangle on...

$AB, AC$ are tangents to a circle and $D$ is the middle point of the chord $BC$. Prove that, if $P$ ...

The vertices of a triangle lie on the lines \[ y=m_1x, \quad y=m_2x, \quad y=m_3x, \] and th...

$AB, AC$ are two given straight lines and $P$ is a given point in their plane. Shew how to draw a li...

Prove that the ortho-centre, the centroid, the centre of the circum-circle, and the centre of the ni...

Prove that the equation of the straight lines bisecting the angles between the lines \[ ax^2+2hxy+...

Prove that the feet of the perpendiculars from any point on the circumcircle of a triangle $ABC$ on ...

Prove that the limiting points of a system of coaxal circles are inverse points with respect to ever...

Prove that if a sphere passes through the eight vertices of a parallelepiped the parallelepiped must...

Any irregular polygon is circumscribed about a circle. Prove that the perimeter of the polygon bears...

Show that the feet of the perpendiculars from a point $P$ on the circumcircle of a triangle lie on a...

Reciprocate with respect to a circle the theorem: From a point $A$ on a circle tangents are drawn to...

Three circles touch one another in pairs. Show that the circle through their points of contact cuts ...

If perpendiculars are drawn from the orthocentre of a triangle $ABC$ on the bisectors of the angle $...

The straight line $x\cos\alpha+y\sin\alpha=p$ being called the line $(\alpha p)$, find the equation ...

If $S$ be the area of a quadrilateral whose sides are $a,b,c,d$, prove that \[ S^2 = (s-a)(s-b)(...

If $r, R$ are the radii of the inscribed and circumscribed circles of the triangle $ABC$ and $s$ the...

Shew that forces represented in all respects by the lines joining any point to the angular points of...

Prove that, if $O, N, H$ are the circumcentre, nine-point centre and orthocentre of a triangle $ABC$...

In any triangle, prove that \[ r = 4R \sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}. \] A po...

In a triangle ABC, D is any point in BC. The angles BAD, CAD, ADC are $\alpha, \beta$ and $\theta$ r...

Express the radii of the inscribed and escribed circles of a triangle in terms of the radius of the ...

From the points of contact of the inscribed circle with the sides of a triangle perpendiculars are l...

The sides of an acute-angled triangle each subtend a right angle at some point not in the plane of t...

(i) Shew that the radii of the circles touching the sides of a triangle are the roots of the equatio...

A plane polygon of $n$ sides of lengths $a_1, a_2, \dots, a_n$, respectively, has angles given by $\...

Shew that the cross-ratio of the pencil $u+\lambda_r v=0$, ($r=1,2,3,4$), is \[ \frac{(\lambda_1-\la...

If $S_r \equiv x^2+y^2+2g_rx+2f_ry+c_r$, interpret geometrically the following equations: \begin{e...

Shew that a quadrilateral with sides of given lengths has its greatest area when it is cyclic. \pa...

$AB$ and $AC$ are two fixed straight lines, and $O$ is a fixed point. Two circles are drawn through ...

If the four faces of a tetrahedron are equal in area, prove that they are equal in all respects....

A circle $C$ has its centre on the circumference of another circle $C'$. Any tangent to $C$ cuts $C'...

$O$ is the circumcentre and $P$ is the orthocentre of a triangle $ABC$. Prove that the resultant of ...

The perimeter and area of a convex pentagon $ABCDE$ which is inscribed in a circle are denoted by $2...

Prove that, in a right-angled triangle, the square described on the hypotenuse is equal to the sum o...

Through each angular point of a tetrahedron a plane is drawn parallel to the opposite face. Prove th...

Prove that the feet of the perpendiculars from any point on the circumcircle of a triangle $ABC$ on ...

Establish the harmonic property of the complete quadrilateral. Given two parallel straight lines...

Prove that if two chords of a circle are perpendicular the tangents at their ends form a quadrilater...

Prove that any line cuts the sides of a triangle in segments the continued product of the ratios of ...

In a tetrahedron show that the perpendicular to any face through its orthocentre intersects all the ...

From a point $P$ on the circumscribing circle of the triangle $ABC$ perpendiculars $PL, PM$ and $PN$...

Prove that the external bisectors of the angles of a triangle meet the opposite sides in collinear p...

Prove that the circle drawn through the middle points of the sides of a triangle also passes through...

The sides of a triangle $ABC$ are cut by a straight line in $D, E, F$. Prove that \[ BD \cdot CE...

In a triangle $ABC$, $D, E$ and $F$ are the middle points of the sides $BC, CA, AB$ respectively and...

Prove that the mid points of the diagonals of a complete quadrilateral are collinear. Any line $L$...

Points $P,Q,R,S$ are taken on the sides $AB,BC,CD,DA$ of a square $ABCD$ such that the figure $PQRS$...

Any point $O$ is taken on the circumcircle of a triangle $ABC$; $X,Y,Z$ are the projections of $O$ o...

Prove that, if opposite edges of a tetrahedron are equal, the line joining the mid-points of any pai...

A rectangle is formed by drawing a pair of parallel lines through two given points A, B and a pair o...

Prove that the mid-points of the six edges of a parallelopiped which do not pass through either of t...

Points $P, Q$ are taken in the sides $AB, AC$ respectively of a triangle $ABC$, so that $AP:AQ :: AC...

Explain how to construct a circle (a) to pass through two given points and to touch a given straight...

Prove that two homographic ranges are mutually projective. $P, Q, R$ are three fixed collinear poin...

The equations of two intersecting straight lines are \[ a_1x+b_1y+c_1=0 \quad \text{and} \quad a_2x...

$A, B, C$ are three given non-collinear points. Prove that three circles can be drawn with $A, B, C$...

Express the radius $R$ of the circumcircle of a triangle $ABC$ in terms of the sides, and prove that...

A mound on a level plane has the form of a portion of a sphere. At the bottom its surface has a slop...

The base $BC$, the angle $A$ and the height of $A$ above $BC$ are given for a triangle $ABC$. Give r...

Prove that $r=R(\cos A+\cos B+\cos C-1)$, where $r, R$ are the radii of the incircle and circumcircl...

Prove that the distance between the orthocentre of a triangle $ABC$ and the centre of the circumscri...

If R and r are the radii of the circumscribed circle and inscribed circle of a triangle ABC, prove t...

While ascending a tower it is found that at a height $a$ from the ground the breadth of a river subt...

Express the area of a triangle in terms of the angles and the radius of the inscribed circle. Prove ...

$I_1, I_2, I_3$ are the centres of the escribed circles of the triangle $ABC$. With the usual notati...

If $O$ and $I$ are the circumcentre and incentre of a triangle $ABC$, show that $OI^2=R^2-2Rr$, wher...

If $a,b,c,d$ are the sides (taken in order) of a quadrilateral inscribed in a circle, prove that the...

Prove that, in areal coordinates, the equation \[ \frac{x}{a}(\frac{y}{b}\cos A - \frac{z}{c}\co...

If $R$ and $r$ are the radii of the circumscribed and inscribed circles of a triangle $ABC$, prove t...

Prove that, if the inscribed circle of a triangle subtends angles $2\theta_1, 2\theta_2, 2\theta_3$ ...

Given a circle of which AB is a diameter, C and D two points on the circumference, find a point P on...

Four forces acting along the sides of a quadrilateral are in equilibrium; prove that the quadrilater...

The tangents to a circle at A and B meet in T, and any line drawn through T cuts the circle in C and...

A statue on a pedestal stands on a slope of inclination $\theta$, and at a certain point on the slop...

Define a coaxal system of circles and its limiting points. Given a coaxal system of circles $S$, pr...

Lines are drawn through the vertices $A, B, C$ of a triangle making angles $\pi/3$ in the same sense...

Prove that the locus of a point $P$, which moves in a plane so that the ratio of its distances from ...

Define the radical axis of two circles and shew how to construct it for two circles which do not int...

Find the radius of the circumcircle of a triangle in terms of the sides. Points are taken on the...

Prove that the three feet of the perpendiculars on the sides of a triangle from any point on its cir...

Prove that the bisectors of an angle of a triangle divide the opposite side into segments whose rati...

Define the centres of similitude of two circles. If a variable circle touches two fixed circles in...

Prove that the curve of intersection of two spheres is a circle. If three circles in space are suc...

The bisector of the angle $BAC$ of a triangle $ABC$ cuts the circumcircle of the triangle in $D$. Pr...

The lines joining the vertices of a triangle $ABC$ to any point $O$ cut the opposite sides in $P,Q,R...

Points $X, Y, Z$ are taken in the sides $BC, CA, AB$ of an equilateral triangle $ABC$ and $AX, BY, C...

Prove that the curve of intersection of two spheres is a circle in a plane perpendicular to the line...

If $a, b, c$ are the sides and $A, B, C$ the angles of a triangle prove, \textit{ab initio}, \be...

Shew that the feet of the perpendiculars on the sides of a triangle from any point on the circumcirc...

Prove that the area of a triangle $ABC$ is $2R^2\sin A\sin B\sin C$, where $R$ is the radius of the ...

If $P$ is the orthocentre of a triangle $ABC$, $O$ the centre of the circumscribing circle, and $R$ ...

Find the equations of the bisectors of the angles between the straight lines \[ ax+by=c \quad \t...

Two straight rulers with inches marked on them are laid across one another at a given angle so that ...

If the bisectors of the angle $A$ of the triangle $ABC$ meet $BC$ in $D,D'$, prove that the radius o...

$ABC$ is a triangle inscribed in a circle whose centre is $O$ and radius $R$; and $AO, BO, CO$ meet ...

The tangents at $B, C$ to the circumcircle of a triangle $ABC$ meet in $L$; $AL$ cuts the circle in ...

$O$ is the circumcentre of a triangle $ABC$ and $AO, BO, CO$ cut the sides $BC, CA, AB$ in $X, Y, Z$...

Prove that the angles made by a tangent to a circle with a chord drawn from the point of contact are...

Let $A,B,C,A',B',C'$ be any six points in space and $O$ any point of the line of intersection of the...

The lines joining the angular points of a triangle $ABC$ to the middle points of the opposite sides ...

Given the inscribed and circumscribed circles of a triangle in position, prove that the orthocentre ...

If $A,B,C,D$ are four points on the same straight line, and circles are drawn through $AB, BC, CD, D...

The surfaces of two spheres have more than one real common point. Prove that they intersect in a cir...

Two regular tetrahedra are formed from among the vertices of a cube of edge length $a$. Find the vol...

The centres of two large solid hemispherical radar domes of radii $a$ and $b$ are at a distance $c$ ...

State Pythagoras's Theorem. Two circles $\alpha$, $\beta$ with centres $A$ and $B$ and radii $a$ and...

A solid fills the region common to two equal circular cylinders whose axes meet at right angles. Pro...

A hill $\frac{1}{2}$ mile high is in the shape of a spherical cap, with a horizontal circular rim, t...

Let $A'$ be a point in the plane of a triangle $BCD$. Let $BC$ and $A'D$ meet at $X$, and $A'B$ meet...

Let $\cal S$ be an infinite set of pairs of points in the plane such that the points in question do ...

A regular tetrahedron, with edges of length $a$, is inscribed in a sphere of radius $R$. Find the va...

$ABCD$ is a tetrahedron. $O$ is a point not lying on any of its faces. The line through $O$ and $A$ ...

Four points $P_1$, $P_2$, $P_3$, $P_4$ are not coplanar. The line through $P_1$ and $P_3$ is denoted...

Take any two of the standard concurrence theorems for the triangle (medians, altitudes, bisectors of...

Find the \emph{width} of a regular tetrahedron of side $a$, where \emph{width} is defined as the lea...

$O$ is a point inside a convex polygon $ABC\ldots N$, of $n$ sides; $A_1, B_1, C_1, \ldots, N_1$ are...

A tetrahedron $ABCD$ is such that there is a sphere which touches its six edges. Prove also that the...

A cube of side $2a$ has horizontal faces $ABCD$, $A'B'C'D'$ and vertical edges $AA'$, $BB'$, $CC'$, ...

Prove that there exists a sphere touching the six edges of the tetrahedron $ABCD$ internally if, and...

From a point $O$ perpendiculars $OA'$, $OB'$, $OC'$, $OD'$ are drawn to the faces of a tetrahedron $...

A convex polyhedron $P$ has, for its faces, $x$ triangles and $y$ (convex) quadrilaterals, where $x$...

Prove that the area of a sphere $S$ between two parallel planes $\pi$, $\pi'$ both of which meet $S$...

A tetrahedron $ABCD$ is given; $L$, $M$, $N$ are the middle points of $BC$, $CA$, $AB$ respectively ...

In a tetrahedron $ABCD$ the edges $AD$ and $BC$ are perpendicular, $AB = CD$, and $AC = BD$. Prove t...

Prove that in general the perpendicular from a vertex on to the opposite face of a tetrahedron is in...

Show that if the distance between the points $A$ and $B$ is greater than $d$, then the two spheres o...

Given any four points on the surface of a sphere of unit radius, prove that it is possible to find t...

Two great circles on a sphere of radius $r$ meet at an angle $A$. Find the areas of the four regions...

If $ABCD$ is a tetrahedron, prove that the lines joining the vertices $A,B,C,D$ to the centroids of ...

A tetrahedron $ABCD$ has edges of lengths $AB=AC=AD=a$, and $BC=CD=DB=b$. A sphere is inscribed in t...

Three circles $A, B, C$ lie in three different planes $\alpha, \beta, \gamma$. The circles $B, C$ me...

In each of the following two cases, either prove the statement true or give a counter-example to sho...

ABCD is a given tetrahedron. A circle in the plane ABC meets BC, CA, AB in the pairs of points $P_1,...

A tetrahedron $ABCD$ has the property that a sphere can be drawn to touch each of its six edges. Pro...

Prove Pappus's theorem that, if $A, B, C$ and $P, Q, R$ are two triads of collinear points on (disti...

$A,B,C,D$ are four points in a plane. Prove that a necessary and sufficient condition for the pairs ...

Determine the relations between the lengths of the edges of a tetrahedron $ABCD$ in order that a sph...

Two spheres have two distinct (real) points in common. Prove that their total intersection consists ...

In a tetrahedron $OABC$ the lengths $OA, OB, OC$ are equal and the angles $BOC, COA, AOB$ are right ...

$A_1A_2A_3A_4$ is a tetrahedron and $O$ is a point in general position. On each edge $A_rA_s$ the po...

A line in space cuts a plane at $P$ and is perpendicular to two distinct lines lying in the plane an...

The circumscribing sphere of a tetrahedron $A_1A_2A_3A_4$ has centre $Q$; $O_1$ is the circumcentre ...

The foot of the perpendicular from a point $O$ to the face $A_2A_3A_4$ of a tetrahedron $A_1A_2A_3A_...

Prove that the tangents drawn to a circle from a given external point are equal. The sides of a skew...

Prove for any tetrahedron that the perpendicular from a vertex on to the opposite face will meet the...

The sides $AB, BC, CD,$ and $DA$ of a skew quadrilateral are cut by a plane in the four points $P, Q...

A man on a hill observes that three vertical towers standing on a horizontal plane subtend equal ang...

If for the segment of a sphere intercepted by a plane, $\lambda$ denotes the ratio of the area of th...

A map of the world is drawn with the parallels of latitude horizontal and the meridians of longitude...

Points $L, M, N$ are taken between vertices on the sides $BC, CA, AB$ respectively of a triangle $AB...

A regular dodecahedron is bounded by twelve regular pentagons each with side of unit length. Prove t...

A fixed point $A$ and a variable point $P$ are taken on a given sphere. $AP$ is produced to $Q$ so t...

The middle points of the edges $AD, BC$ of a tetrahedron $ABCD$ are $L, M$ respectively, and $P$ is ...

On the surface of a sphere, centre $O$, are four points $A, B, C, D$. Prove that $AB$ is perpendicul...

Prove that there are five, and only five, types of regular polyhedrons. Calculate the number of edge...

$ABCD$ is a tetrahedron, and $H_1$ and $H_2$ are the orthocentres of the triangles $BCD, CAD$ respec...

AC and BD are two skew lines in space. A plane meets AB at P, BC at Q, CD at R and DA at S. Prove th...

A rectangle $R$ has centre $M$ and sides $2a, 2b$. A point $O$ is taken on the line through $M$ perp...

Solve: \begin{align*} x\cos\alpha + y\cos\beta + z\cos\gamma &= 1, \\ x\sin\alpha + y\sin\beta +...

A torus is the figure formed by rotating a circle of radius $a$ about a line in its own plane at a d...

A roof whose slope is inclined at $30^\circ$ to the horizontal runs into another roof whose slope is...

$A$ is a variable point $(X,0)$, $P$ and $Q$ are points $(h, k)$, $(h', k')$, respectively. Show tha...

The two parabolas \[ y^2 = 4ax, \quad y^2=4bx, \] are drawn, where $a$ and $b$ are both posi...

Obtain the equation of a tangent to the parabola $l/r = 1+\cos\theta$ in the form $l/r = \cos\theta ...

Prove that the two circles $3x^2+3y^2+6ax = a^2$ and the hyperbola $6x^2 - 3y^2 = 2a^2$ are so relat...

Shew that the equation of the normal at a point $(\alpha, \beta)$ of the curve $f(x,y)=0$ is \[ (x-\...

Show that an infinity of straight lines can be drawn to meet three given straight lines $a, b, c$ in...

Express the area $S$ of a triangle in terms of the lengths of the sides. \par Prove that \[ ...

Find the ratio of the volume of a regular tetrahedron to the volume of the regular tetrahedron forme...

The top $M$ of a mountain is observed from the ends $A, B$ of a base of length 4000 yards. The compa...

Calculate the volume common to two spheres, each of radius $a$, which are so placed that the centre ...

Give an account of the method of Orthogonal Projection with illustrations of its use. Consider the f...

Prove that the tangents from any point to a sphere generate a circular cone. A variable small ci...

A wireless signal from an aeroplane is intercepted at two direction-finding stations $A$ and $B$ whi...

Determine the centre and the radius of the circle inscribed in the triangle formed by the lines $3x+...

Obtain the equation of the circumcircle of the triangle of reference in areal coordinates $(x, y, z)...

Prove that three straight lines in space, parallel to the same plane but not to one another, can be ...

If $(x,y,z)$ are the homogeneous coordinates (e.g. areal or trilinear coordinates) of a point in a p...

Show that in general two spheres can be inscribed in a right circular cone to touch a given plane no...

Points $D, E$, and $F$ are taken on the sides $BC, CA,$ and $AB$, respectively, of the triangle $ABC...

Shew that there are two spheres which touch a given right circular cone along circles and also touch...

Find the equation of the line $l$ joining the points of intersection of $x = \lambda y$ and $x = \mu...

If $A, B, C, D$ are any four coplanar points, prove that the three pairs of lines through any point ...

(i) Find the locus of the feet of the perpendiculars from a fixed point $O$ to the straight lines wh...

Prove that the equations of the sides of a quadrilateral may, by a suitable choice of the triangle o...

P, Q and R are corresponding points of homographic ranges on three lines p, q and r which do not lie...

XYZ is the triangle of reference and H, K are the points $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$. The l...

Show that two circles, in different planes, which have two points common lie on a sphere. A tetr...

Prove that the equation of the circumcircle of the triangle whose sides lie along the lines $ax^2+2h...

If P, Q, R are three points with homogeneous coordinates $(p, g, h), (f, q, h), (f, g, r)$, respecti...

Show that through any point in space one line can be drawn to meet each of two other lines which do ...

Shew that there are two spheres, real, coincident, or imaginary, which pass through three given poin...

The sum of the lengths of the twelve edges of a rectangular box is $5l$, and the sum of the areas of...

In a system of generalized homogeneous coordinates $(x,y,z)$ the condition that the lines $lx+my+nz=...

A family of conics touching the sides of a given triangle have their axes parallel to a given straig...

Find in trilinear coordinates the equation of the circle which has for its diameter the perpendicula...

Find the condition that the equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] may represent two straight lin...

Define the polar of a point with respect to a circle and show that a straight line through a point c...

$ABCD$ is a tetrahedron with a fixed base triangle $ABC$ and a variable apex $D$. Shew that the perp...

$AB$ is a diameter of a circle whose centre is $O$; $ODC$ and $BEC$ are straight lines cutting the c...

$ABCD$ is a horizontal line and $DE$ a vertical line. $DE$ subtends angles $\theta, 2\theta, 3\theta...

If every edge of a tetrahedron is perpendicular to the edge that it does not meet, prove that the pe...

Prove that the three lines, each of which forms a harmonic pencil with the three lines \[ y=0, \quad...

If $O, D, E$ and $F$ are the centres of the inscribed and escribed circles of a triangle, prove that...

Two points $H(1,1,1)$ and $H'(p,q,r)$ are taken in the plane of the triangle of reference $ABC$. $AH...

The straight line \[ l \equiv \alpha x + \beta y + \gamma z = 0 \] meets the sides $BC, CA, ...

$XYZ$ is the triangle of reference and $P$ is the point $(f,g,h)$. The line $XP$ meets $YZ$ in $L$, ...

Given two vertices of a triangle and its area, shew that the locus of its orthocentre is two parabol...

Prove that of the circles \begin{align*} b(x^2+y^2) + a^2(2y-b) &= 0, \\ a(x^2+y^2) + b^2(...

Prove that the centre of the circle inscribed in the triangle formed by the external common tangents...

The coordinates of two points $P(p',0,p)$ and $Q(q',q,0)$ on the sides $y=0$ and $z=0$ of the triang...

Shew that if the sum of the squares on a pair of opposite edges of a tetrahedron is equal to the sum...

Prove that the equation $S=x^2+y^2+2gx+2fy+C=0$ represents a circle, and examine the meaning of $S$ ...

Define the radical axis of two circles. Given two circles $A, B$ and a straight line $L$, draw a cir...

Show that if the sides of the pedal triangle of the triangle $ABC$ be produced to meet the opposite ...

Prove that the general equation of a circle in areal coordinates is \[ (x+y+z)(t_1^2x+t_2^2y+t_3^2z)...

A light elastic string of modulus $\lambda$ and natural length $l$ has its ends attached to two fixe...

Prove that the equations \[ \frac{x}{5-6t-3t^2} = \frac{y}{5+8t-t^2} = \frac{1}{1+t^2} \] re...

$V$ is the middle point of a given chord $AB$ of a given circle. $PQ$ is any parallel chord. $QV$ me...

$PQ$ is a chord of a parabola that passes through the focus $S$. Two circles are drawn through $S$, ...

Any number of spheres touch a plane at the same point $O$. Prove that any plane, not through $O$, cu...

$P$ is any point within a triangle $ABC$, and at a distance $d$ from its circumcentre, the circumrad...

Prove that in areal coordinates the equation of the circumcircle of the triangle of reference is $a^...

Prove that the lines joining the middle points of pairs of opposite edges of a tetrahedron are concu...

Prove that the sum of all the plane angles forming any solid angle is less than four right angles. ...

Prove that any two lines in space are cut proportionately by three parallel planes. AB is the co...

Define a system of coaxal circles. Prove that one circle of the system can be drawn through any give...

Three vertical flagstaffs stand on a horizontal plane. At each of the points $A, B$ and $C$ in the h...

An observer looking up the line of greatest slope of an inclined plane sees a vertical tower due Eas...

Prove that the inverse of a sphere with respect to any internal point is a sphere. Invert with respe...

$ax+by+c=0$ is one asymptote of a hyperbola which passes through the origin and which touches the st...

Shew that if three of the four perpendiculars from the vertices of a tetrahedron on to the opposite ...

State the condition that the equation $ax^2+by^2+2hxy+2gx+2fy+c=0$ shall represent two straight line...

$ABCD$ is a quadrilateral circumscribing a circle and $a,b,c,d$ are the lengths of the tangents from...

A uniform rod $PQ$, of length $l$, rests with one end $P$ on a smooth fixed elliptic arc whose major...

Find the conditions that \begin{enumerate} \item[(i)] $ax^2+2hxy+by^2$, \item[(i...

Give an account of the method of reciprocation with respect to a circle, and illustrate its use....

Shew that a pencil of four rays cuts any transversal in a range of constant anharmonic ratio. Ex...

Prove that the focus of a parabola which touches the sides of a triangle lies on the circumscribing ...

Show that there is in general one circle of a coaxal system which cuts a given circle orthogonally. ...

$A, B, C, D$ are the corners of a square of side $a$ on level ground. Inside the square is a flagsta...

Find a formula for the radius of the inscribed circle of a triangle. The circle inscribed in the...

Shew that, by a proper choice of axes, the equations of any two circles may be written in the form $...

Any point $X$ is taken in the side $CD$ of a rectangle $ABCD$, and the line through $A$ perpendicula...

Prove that, if a straight line be at right angles to two intersecting straight lines, it will be at ...

Prove that the locus of points whose tangents to the two conics \[ S = ax^2+by^2+cz^2=0, \quad S...

Prove the formula \[ \frac{(s')^6}{\rho^2\sigma} = \begin{vmatrix} x' & y' & z' \\ x'' & y'' & z...

The equations of two circles in space are \begin{align*} 2x+2y-z=0, &\quad 5x^2+5y^2+8z^...

Three infinite parallel wires cut a plane perpendicular to them in the angular points $X,Y,O$ of an ...

Shew how to draw a perpendicular to a plane from a point outside it. Prove that if two straight li...

Prove that there are six normals from a point $(f,g,h)$ to the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}...

Shew that the equation of the osculating plane at the point $(x,y,z)$ of the sphero-conic in which t...

Show that a right circular cone can be drawn to touch three consecutive osculating planes of a curve...

Investigate the two dimensional motion of an incompressible fluid defined by the stream function $\p...

Explain carefully how the azimuth of the sun at any given time at a known point on the earth's surfa...

$ABCD$ is a square, whose opposite vertices $A,C$ lie, respectively, on the lines $y = mx, y = -mx$....

Let $n$ be a positive integer. What is the largest number $M$ of maxima that the polynomial \[f(x) =...

A farmer wishes to provide his cattle with three nutrients $A, B$ and $C$, for which the minimum req...

Discover all the real roots of each of the equations \begin{enumerate} \item[(i)] $(x-1)^3 + (x-2)^3...

Let $J_1$ be the operation of taking the inverse (reciprocal) of a number, and $J_2$ the operation o...

Determine $\theta$ so that the line \[lx + my + n = \theta(l'x + m'y + n')\] is perpendicular to the...

(i) Show that in rectangular cartesian coordinates the equation $$p(x^4 + y^4) + qxy(x^2 - y^2) + rx...

A right-angled triangle has integral sides and the lengths of the two shorter sides differ by 1. If ...

If $x+y+z+t=0$, prove that \begin{enumerate} \item[(i)] $(x^3+y^3+z^3+t^3)^2 = 9(xyz+yzt...

If $a, b, c$ are unequal non-zero numbers, solve the simultaneous equations \begin{align*} ...

Solve the equations \begin{align*} x+y+z &= 3, \\ x^2+y^2+z^2+2z &= 9, \\ xyz+xy &= -2. ...

Prove that \[ a^3+b^3+c^3-3abc = \tfrac{1}{2}(a+b+c)[(b-c)^2+(c-a)^2+(a-b)^2]. \] Hence, or otherwis...

Solve: \begin{align*} y^2+yz+z^2 &= 1, \\ z^2+zx+x^2 &= 4, \\ x^2+xy+y^2 &= 7. \end{align*}...

Prove that if the internal energy of a certain gas is a function of the temperature only, and its pr...

Prove that if \[ a\frac{y+z}{y-z} = b\frac{z+x}{z-x} = c\frac{x+y}{x-y}, \] each of these expressi...

Solve the equations \[ \frac{1}{y} - \frac{1}{z} = a - \frac{1}{a}, \quad y - \frac{1}{z} = b - ...

Show that if $ax+by+cz=0$ for all values of $x, y,$ and $z$ such that $\alpha x + \beta y + \gamma z...

Having given that \[ \frac{x^2-yz}{a} = \frac{y^2-zx}{b} = \frac{z^2-xy}{c}, \] prove that ...

If $l, m, l', m', l''$ and $m''$ are integers, and if $\alpha/\beta$ is not rational, and if \[ l\a...

Prove that, if $p/q$ is a fraction in its lowest terms, then integers $r$ and $s$ can be found such ...

At an election the majority was 1184, which was one-fifth of the total number of votes; how many vot...

An article when sold at a profit of 13 per cent. yields 1s. 5d. more profit than when sold at a prof...

A figure of four triangles and three squares is constructed by describing squares P, Q, R externally...

If \begin{align*} a(y^2+z^2-x^2) &= b(z^2+x^2-y^2) = c(x^2+y^2-z^2), \\ \text{and } x(b^...

Find all the real solutions of the equations: \begin{enumerate} \item[(i)] $x(x^2+y^2)=6y, \qu...

If the coordinates of any point referred to two different sets of axes (not necessarily rectangular)...

Find rationalising factors for the expressions \begin{enumerate} \item $x^{2/3} + x^{1/3...

Prove that a point and its inverse with respect to a circle $C$ invert into a point and its inverse ...

A bowl is in the shape of a segment of a sphere, greater than a hemisphere. The diameter of the hori...

Shew that, if \[ \frac{x^2}{a} + \frac{y^2}{b} = x+y \quad \text{and} \quad \frac{a^2}{x} + \fra...

Prove that \[ \begin{vmatrix} \sin 2A & \sin C & \sin B \\ \sin C & \sin 2B & \sin A \\ \sin B &...

Prove that if \[ y^2+z^2+yz=a^2, \quad z^2+zx+x^2=b^2, \quad x^2+xy+y^2=c^2, \quad yz+zx+xy=0, \] ...

Find the angle between the two straight lines \[ ax^2+2hxy+by^2=0. \] Prove that the equation of the...

Simplify the expression \[ \frac{\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{(x+y)^2}{(a+b)^2}}{\frac{...

Shew that the area bounded by the parabola $ay=x^2$ and the lines $y=x, y=2x$ is $\frac{7}{6}a^2$....

If $a+b+c=0$ and $x+y+z=0$, prove that \[ a^2x^2+b^2y^2+c^2z^2-bcyz-cazx-abxy = \frac{1}{4}(a^2+...

Solve the equation \[ \frac{1}{\sqrt{a+x}-\sqrt{a}} + \frac{1}{\sqrt{a+x}+\sqrt{a}} = \frac{m}{\...

If the sum of two positive numbers is given, prove that their product is greatest when they are equa...

Solve the equations: \begin{enumerate} \item[(i)] $x^2+x+1 = \sqrt{x}(2\sqrt{x}-1)(x+\sq...

Find, to the nearest penny, the difference between the simple and the compound interest on £350 for ...

Factorise: \begin{enumerate}[(i)] \item $a^3+2a-(b^3+2b)$, \item $2xy+y^2-z^2+2x...

Shew that the equation $axy+bx+cy+d=0$ may be written in the form \[ \frac{x-p}{x-q} = \lambda \fr...

Prove that the geometric mean of $n$ positive numbers cannot exceed their arithmetic mean. Deduce th...

Positive rational 'weights' $m_1, \ldots, m_n$ are attached to positive numbers $a_1, \ldots, a_n$. ...

Prove that, if $n > 1$, $$\sum_{r=1}^n \left(1 + \frac{1}{r}\right) > n(n+1)^{1/n}.$$ If you appeal ...

Prove that (if all the numbers involved are positive) $$(ab)^{\frac{1}{2}} \leq \frac{1}{2}(a+b) \qu...

$a$, $b$, $c$ are three positive numbers. Prove the inequality $$abc \geq (b + c - a)(c + a - b)(a +...

Prove that the geometric mean of a finite set of positive real numbers does not exceed their arithme...

Prove that if $x_1, x_2, \ldots, x_n$ and $y_1, y_2, \ldots, y_n$ are two sets of positive quantitie...

Let $p_i$ ($1 \leq i \leq n$) and $q_i$ ($1 \leq i \leq n$) be real numbers such that $$p_1 \geq p_2...

Prove that, if $ax^2+2bx+c$ is to be positive for all real values of $x$, it is both necessary and s...

Two polynomials, $P$ and $Q$, have no factor in common. Shew that the maximum and minimum values of ...

Find the condition that $ax+b/x$ can take any real value for real values of $x$. Express $\xi = (x-a...

Shew that two quadratic expressions $ax^2+2bx+c$ and $a'x^2+2b'x+c'$ can generally be expressed in t...

Shew how to express $ax^2+2bx+c$ and $a'x^2+2b'x+c'$ simultaneously in the forms $p(x-\alpha)^2+q(x-...

If $x, y, z$ be real, prove that \[ a^2(x-y)(x-z)+b^2(y-x)(y-z)+c^2(z-x)(z-y) \] is always p...

Simplify $\dfrac{2x+5}{6x+7} - \dfrac{2x-1}{6x+5} - \dfrac{32x+33}{36(x+1)^2-1}$. and prove that...

Obtain the condition for the equation $ax^2 + 2bx + c = 0$ to have real roots, where $a$, $b$ and $c...

Let $f(x) = ax^2 + bx + c$ ($a$, $b$, $c$ real, $a > 0$). Explain why the following statements are e...

Express $(a^2+b^2+c^2)(x^2+\beta^2+\gamma^2)-(a\alpha+b\beta+c\gamma)^2$ as the sum of three squares...

Let $a, b, c$ be integers and let $f(x, y) = ax^2 + 2bxy + cy^2$. Show that there are integers $p, q...

Prove that, if $a$, $b$, $h$ are real numbers such that $a > 0$, $ab - h^2 > 0$, then \[ax^2 + 2hx +...

If \[x_1x_2 + y_1y_2 = a_1, \quad x_2x_3 + y_2y_3 = a_2, \quad x_3x_1 + y_3y_1 = a_3, \quad x_1^2 + ...

Determine the limitations, if any, on the value of $p$ if the expression $$x^2(y^2 + 2y + 2) + 2x(y^...

Prove that the expression \[5x^2 + 6y^2 + 7z^2 + 2yz + 4zx + 10xy\] is positive for all real values ...

What conditions on the real numbers $a$, $b$, $c$ are needed to ensure that \begin{align} \frac{ax^2...

Prove that, if $a>0$ and $ac-b^2>0$, then $ax^2+2bx+c > 0$ for all real values of $x$. Examine wheth...

If $\alpha, \beta$ denote the roots of a given quadratic equation $Ax^2+Bx+C=0$, find the quadratic ...

A train travels 525 miles; if its average rate had been $2\frac{1}{2}$ miles per hour faster, it wou...

Find the conditions that the equation $ax^2+2bx+c=0$ should have (i) both its roots positive and (ii...

Given that \[xy - 3x - 2y + 4 = 0,\] evaluate \[\frac{(x-1)(y-4)}{(x-4)(y-1)}.\] If also \[xz - 6x -...

Solve the equations \begin{align} x + y^3 + z^3 &= 0,\\ x^3 + y + z^3 &= 0,\\ x^3 + y^3 + z &= 0, \e...

Prove that, if $x_1$ and $x_2$ are connected by the relation $$ax_1x_2 + bx_1 + cx_2 + d = 0,$$ ther...

If $x$, $y$, $z$ are all different and $x + \frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x},$ prove ...

Given that $s^2 + c^2 = 1$, prove that \[(4c^3 - 3c)^2 + (4s^3 - 3s)^2 = 1.\] Given, conversely, tha...

Show that, when $a,b$ and $c$ are real and positive, the system of equations \begin{equation} \tag{1...

Solve the equations \[ x (x-a) = yz, \quad y(y-b) = zx, \quad z(z-c) = xy. \]...

Express the coordinates of points of the cubic curve $y^2=x^2(1+x)$ in terms of a parameter $t$ by p...

Solve the equations \begin{enumerate} \item $\frac{2x}{x-a} + \frac{x}{x-b} = 3$. ...

Solve the equation \begin{enumerate} \item[(1)] $\frac{(x-1)^3}{(x+2)^3} = \frac{x-4}{x+5}$. ...

If \[ x(1+\sin^2\phi-\cos\phi) = (y\sin\phi+a)(1+\cos\phi) \] and \[ y(1+\cos^2\phi) = (...

Solve the equations: \begin{enumerate}[(i)] \item $x^2y^2 - 25xy + x^2+y^2+1=0, \quad xy...

Solve the equations: \begin{enumerate} \item[(i)] $\sqrt{x^2-40x+39}=0$; \item[(...

Solve the equations: \begin{enumerate} \item[(i)] $u+v=2, \quad ux+vy=-1, \quad ux^2+vy^...

Solve for $x, y, z$ in terms of $p, q, r$ the simultaneous equations \begin{align*} x+y+...

Solve the equations: \begin{enumerate}[(i)] \item $(3x-1)^2+48x=16$, \item $\dfr...

Find the equation of the tangent at a point on the curve given by \[ x/t = y/t^2 = 3a/(1+t^3). \] ...

Show that, for every positive integer $n$, the number $n^9 - n$ is divisible by 30 and that, for eve...

The letters $n$ and $k$ denote positive integers. \begin{enumerate} \item[(i)] Show that $n^3 - n^3$...

Suppose that $n$, $x$ and $y$ are positive integers such that $n+x$ is a square and $n+y$ is the nex...

Show that if $a, b, c, d \in \mathbb{Q}$, the rational numbers, and $a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}...

Show that if $a$, $b$, $c$ are integers it is always possible to find integers $A$, $B$, $C$ such th...

If $p$, $q$, $r$, $s$ are positive integers with $qr - ps = 1$, prove that any fraction which lies b...

The number $n$ whose digits in the scale of 10 are $a$, $b$, $c$, $d$ in that order is the same as t...

$a, b, c, d$ are integers lying between 1 and 9, inclusive, and $$n = 10^4a + 10^3b + 10^2b + 10c + ...

Prove that the Arithmetic Mean of a number of positive quantities is never less than their Geometric...

Shew that the number of divisors (unity and the number itself included) of the number \[ N = p_1...

Prove that if $a, b, c$ are in arithmetical progression, and $a, b, d$ in harmonical progression, th...

State any rules you know for determining whether a number is divisible by 2, 3, 4, 5, 8, 9, and 11. ...

Prove that the product of any set of integers, each of which can be expressed as the sum of the squa...

Prove that the arithmetic mean of $n$ positive numbers which are not all equal exceeds their geometr...

If $a, b, c$ and $d$ are all real, and if $(a^2+b^2+c^2)(b^2+c^2+d^2) = (ab+bc+cd)^2$, prove that $a...

Find the smallest positive integer which, when divided by 28, leaves a remainder 21, and when divide...

If $A=a^2(a+b+c)+3abc$, $B=b^2(a+b+c)+3abc$ and $C=c^2(a+b+c)+3abc$, where $ab+bc+ca=0$, then $(AB+B...

Explain the method of finding positive integral values of $x$ and $y$ which satisfy the equation $ax...

The function $\mu(n)$ is defined as being equal to 0 when $n$ contains any squared factor, to 1 when...

A 3-inch square tile is decorated by dividing one face into 9 equal squares, and painting the result...

A party of seven people arrives at a tavern which has six vacant rooms. In how many ways can they be...

(i) Prove that 24 is the largest integer divisible by the product of all integers less than its squa...

A theorem in combinatorial theory may be stated as follows: Let $G_1, G_2, ..., G_n$ be $n$ girls an...

Show that there are less than 300 primes $p$ with $1000 \leq p \leq 2000$....

Let $\xi$ be any irrational number. Show that, given any integer $a$, there is a unique integer $b$ ...

Suppose that of the 6 people at a party at least two out of every three know each other, and that al...

Let $x$ be a positive non-zero integer. $S^1(x)$ will denote the sum of the digits of $x$ when writt...

Find all positive integers that are equal to the sum of the squares of their digits....

Six equal rods are joined together to form a regular tetrahedron. Two scorpions are placed at the mi...

Let $N = p_1^{a_1}p_2^{a_2}\ldots p_n^{a_n}$ be the representation of $N$ as a product of powers of ...

A spacecraft may be regarded as a solid body which is convex (i.e. no straight line meets its surfac...

A finite number of circles, not intersecting or touching each other, are drawn on the surface of a s...

The sequence $a_0, a_1, \ldots, a_{n-1}$ is such that, for each $i$ $(0 \leq i \leq n-1)$, $a_i$ is ...

The one-player game of Topswaps is played as follows. The player holds a pack of $n$ cards, numbered...

In a tournament everybody played against everybody else exactly once, and no game ended in a draw. S...

The bus routes in a town have the following properties. \begin{enumerate} \item[(i)] Any two bus sto...

A convex polyhedron is such that precisely three faces concur in each vertex, and that every face is...

5 points lie within a unit square, or on its boundary. Prove that some pair of them are at a distanc...

On a chess board, which consists of 64 squares, a bishop is only allowed to move diagonally. In orde...

A triangle is called chromatic if all its sides are the same colour. Each pair of $n$ distinct point...

Let $d_1, d_2, ..., d_k$ be the distinct positive divisors of the positive integer $n$, including 1 ...

Show that $n$ coplanar lines in 'general position' (i.e. no two lines parallel, no three lines concu...

For any real number $x$, $[x]$ denotes the greatest integer not exceeding $x$. Evaluate, for positiv...

A convex solid bounded by triangular faces is such that, at each vertex, either three or four edges ...

Show that three distinct points in the plane with integral coordinates (in the usual Cartesian syste...

A circular table has radius 1 ft. Five equal circular discs are symmetrically placed so as to cover ...

Let $N_+$, $N_-$ be the number of positive integers of the form $3k + 1$, $3k - 1$, respectively, wi...

Prove that for each positive integer $n$ there is a positive integer $m$ such that the decimal repre...

Five schools play a rugby football competition, each school playing each of the others twice, once a...

A regular octahedron is oriented by assigning a direction along each edge, in such a way that the bo...

A stream of particles moving at speed $v$ falls upon a perfectly elastic plane reflecting surface at...

Forces of 1, 2, 3, 4, 5, 6 pounds weight respectively act at the corners of a regular hexagon inscri...

Three equal uniform rods $OA, OB, OC$ freely jointed at $O$ form a tripod with the feet $A, B, C$ sy...

Eliminate $x, y, z$ from the equations \begin{align*} (z + x - y) (x + y - z) &= ayz, \\...

A circular iron plate conducts 15 Pound-Centigrade thermal units per minute through its thickness pe...

Two bar magnets are each of length 50 cm., but their pole strengths are 100 and 50 units respectivel...

From a stretch of level country, the ground rises at a steady slope of 1 in 30. A railway cutting ru...

For a carbon filament electric lamp, a portion of the curve connecting P.D. and current is found to ...

Shew that the electrical resistance between opposite corners of a framework of twelve equal wires ar...

A man is 2 miles from the nearest point $A$ of a straight road, and he wishes to reach a point $B$ o...

Prove that, if $a$ and $b$ are positive integers ($a < b$), the proper fraction $a/b$ can be expres...

Prove that the sum of the reciprocals of all positive integers which can be written (in the ordinary...

Three collinear points $A, B, C$ are given. Give a construction, making use of a ruler only, for the...

A system of curves is given by the equation $f(x,y,c) = 0$, where $c$ is a variable parameter. Show ...

It is required to place forces in the sides of a given plane quadrilateral so that they shall have a...

The diagram represents a roof truss composed of seven equal bars $AE, EC, CD, DB, EF, FD, CF$ and tw...

$A$ and $B$ are two pegs on the same horizontal and at distance $d$ apart. A square picture frame of...

Forces $P\cos A, P\cos B, P\cos C$ act along the sides $CB, AC, AB$ of a triangle $ABC$, in the dire...

A weight $W$ is attached to a fixed point by four light strings. At the mid-point of each string, wh...

Shew graphically or otherwise that the equation $10^{x-1} = 2x$ has only two real roots and by means...

Obtain the real solutions of the equations \[ x^3 + \frac{7}{3}xy^2 = y^3 + \frac{7}{3}yx^2 = 1 \]...

Prove that the least positive root of the equation \[ x = 2\pi \sec x \] is $2\pi$, and that the...

Give an account of the methods employed for the solution of triangles, giving as many alternative me...

A chord of the curve $y=f(x)$, parallel and near to the tangent at the point $P(\xi, \eta)$, meets t...

Give a short account of the method of generalisation by projection. Obtain the projective generalisa...

Two coplanar triangles $ABC$ and $A'B'C'$ are in perspective from a point $O$. Prove that, of the ni...

Points L, M are taken on the sides AB, AC, respectively, of a triangle ABC so that $BL = \lambda.BA$...

The 1000 yards range trajectory of a rifle bullet is given by the following heights (in feet) above ...

A circular hill is very nearly of the form given by a regular truncated cone 3000 ft. in diameter at...

Explain briefly the principle of virtual work. A frame to form a girder consists of 19 rods of e...

Write an account of the theory of plane frames formed of light rigid bars, freely jointed, consideri...

Shew how to determine graphically the resultant of a system of given coplanar but non-concurrent for...

A semicircular track is made on a hillside, which is inclined at 20° to the horizontal, so that the ...

Explain the construction of the funicular polygon, showing in particular what it becomes when the sy...

The velocity of a stream between parallel banks at distance $2a$ apart is zero at the edges and incr...

$ABE$ is an isosceles triangle, right angled at $A$. $BCDE$ is a square on the opposite side of $BE$...

A train slows down on entering a station and stops with a slight jerk. Discuss the motion of a slidi...

A particle $P$ is attracted towards each of four points $A, B, C, D$ by forces equal to $\mu_1 PA, \...

You are given a number of unequal particles and a number of unequal pieces of elastic string. Explai...

A weight $3w$ is supported by a tripod standing on the ground. Each leg of the tripod is of length $...

ABCD, A'BC'D' are crossed light rods pivoted at B; \[ AB = A'B = 1\frac{1}{2}\text{ ft.},\quad B...

Solve the equations \[ \frac{x^2+y^2+z^2-a^2}{x} = \frac{x^2+y^2+z^2-b^2}{y} = \frac{x^2+y^2+z^2...

In solving a triangle in which two sides and the included angle are given, shew how to determine the...

Show that any surd can be converted into a continued fraction and prove that if $a$ is positive \[...

In order to obtain the height $z$ of an aeroplane above the horizontal plane of a triangle $ABC$ its...

The side $a$ and angle $A$ of the triangle $ABC$, whose area is $\Delta$, are constant. Shew that, w...

Shew that, if $n$ is a positive integer, the number of solutions of the equation \[ n = 2n_1 + 3n_2...

Prove that for any triangle $ABC$, and a point $D$, a point $D'$ may be found such that $DD'$ subten...

Four points $A, B, C, D$ are marked on a straight line so that $AB=14''$, $AC=7''$, $AD=6''$. Shew t...

Two fixed lines which do not intersect are taken in space: shew that in a definite direction one and...

Two circles lie in different planes: prove that in general four circles can be drawn to touch both c...

Two points $P(x,y)$ and $P'(x',y')$ in a plane are said to correspond, when their co-ordinates are c...

Shew that if two coplanar triangles are in perspective from a point, called the centre of perspectiv...

Shew that if $A, B, C$ and $A', B', C'$ are sets of points on two coplanar lines, then the points of...

$D$ is a point in the base $BC$ of a triangle $ABC$, and a line through $D$ meets $AB$ and $AC$ in $...

$A$ and $B$ are two fixed points on a fixed circle. $PQ$ and $P'Q'$ are a variable pair of chords pa...

$P$ is any point on the circumcircle of a triangle $ABC$ and $A', B', C'$ are the other ends of the ...

The lines joining the vertices $A, B, C$ of a triangle to a point $P$ cut the opposite sides in $L, ...

A variable obtuse-angled triangle inscribed in a fixed circle with centre $O$ has a fixed orthocentr...

Construct a triangle of which the sides are bisected at three given points. Prove that it is a d...

Prove the algebraic theorem that, if the product of $n$ positive factors has an assigned value $C$, ...

Rationalise the equation $\sqrt{x}+\sqrt{y}+\sqrt{z}+\sqrt{t}=0$, and express the result in factors ...

Prove that there are four lines in a plane the respective shortest distances of which from three fix...

Three spherical balls, two of which have a radius of 1 inch and the third a radius of 2 inches, rest...

Through a point $K$ inside a triangle $ABC$ a line $XX'$ is drawn parallel to $BC$ to meet the other...

Determine the potential energy of a stretched string. A uniform elastic ring rests horizontally on a...

A plane quadrilateral is formed by the four straight lines $l_i$ ($i=1,2,3,4$), and the point of int...

A manufacturer's expenses are a fixed sum together with a fixed amount $c$ for each article sold. Th...

An equilateral triangle is constructed with its angular points on the sides respectively of the tria...

Solve the equations: \begin{enumerate} \item $\frac{(x-1)^3}{16} - \frac{(x-2)^3}{125} =...

If $a, b, c$ are the sides of a triangle and $2s$ is their sum, prove that the area of the triangle ...

If a chord of a circle passes through a fixed point within the circle, the rectangle contained by it...

The angular points of a rectangle A, B, C, D are the middle points of the sides of a plane quadrilat...

Solve the simultaneous equations: \begin{align*} x(y+z-x) &= a^2, \\ y(z+x-y) &= b^2, \\ ...

\begin{enumerate} \item Show that if $a_1+a_2+\dots$ be a divergent series of positive terms...

Find the limits of \begin{enumerate} \item $n\{e-(1+\frac{1}{n})^n\}$; \quad (ii) $n\lef...

A family of conics circumscribe the triangle $ABC$ and pass through its centroid $G$. Tangents to on...

Construct a circle which shall bisect the circumferences of three given circles....

Find four consecutive numbers which are divisible by 5, 7, 9, 11 respectively....

The straight lines $AB$ and $CD$ intersect in $U$. $AC$ and $BD$ in $V$; $UV$ intersects $AD$ and $B...

Two circles $S, S'$ meet in $A$ and $B$, and the centre $O$ of $S$ lies on the circumference of $S'$...

If $P_1, P_2, \dots, P_n$ and $Q_1, Q_2, \dots, Q_n$ are two homographically related ranges on the s...

Pappus's theorem states that if $A, B, C$ and $A', B', C'$ are two sets of three collinear points in...

A parabola $S$ touches the sides $BC, CA, AB$ of a triangle $ABC$ at $L, M$ and $N$. $BM$ meets $CN$...

Prove that the circles which circumscribe the four triangles formed by four straight lines have a co...

Prove that, if $P, A, B, C$ are four points in a plane, there is another point $P'$ in the plane suc...

Prove by induction or otherwise that if $r$ is a positive integer then the sum of the infinite serie...

Taking the distance of the Sun to be 93,000,000 miles, compare the gravitational effect of the Sun a...

Shew that, if $a, b, c, x, y, z$ denote real numbers, and the sum of any two of the three $a, b, c$ ...

Shew that if \[ y^2+yz+z^2 = a^2, \quad z^2+zx+x^2=b^2, \quad x^2+xy+y^2=c^2, \] then $x+y+z...

Any number of forces $P_1, P_2, \dots, P_n$ in the same plane are in equilibrium. The direction of e...

$A$ is the vertex and $P$ any other point on a uniform catenary. The normals to the catenary at $A$ ...

Shew that the Arithmetic mean of a number of positive quantities is never less than their Geometric ...

If $a,b$ and $c$ are all positive, show that \[ 3(a^3+b^3+c^3) \ge (a^2+b^2+c^2)(a+b+c). \] ...

Five points in a plane are given, no three of them lying on a straight line. Prove that at least one...

The diagram represents a girder bridge in which the horizontal and vertical girders are of equal len...

Solve the equations: \[ x^2+y+z = y^2+z+x = z^2+x+y = 3. \] Eliminate $x,y,z$ from the equat...

The surface bounded by the parabola $x^2=4ay$, the axis of $y$ and the line joining the points $(0,h...

Two circles intersect in $P$ and $Q$. Draw a straight line through $P$ so that the segments of the l...

It is required to find two numbers, each of two digits, such that the first number is equal to the p...

A point $P$ is taken within a triangle $ABC$, whose sides are $a, b, c$, such that $\frac{AP}{a} = \...

Find the area of a triangle, the coordinates of whose angular points are given. $A, B, C, D$ are...

Shew how to solve a triangle $ABC$ having given $B-C, b-c$ and the perpendicular distance of $A$ fro...

Prove that the arithmetic mean of $n$ positive quantities is not less than their geometric mean. ...

Solve the equations \begin{enumerate} \item[(i)] $ax^2+by^2+cz^2=1$, $lx+my+nz=0$, $l'x+...

A gun is fired from a fort $A$, and the intervals between seeing the flash and hearing the report at...

Two points $A, B$ in space are on the same side of a plane. Find a point $P$ in the plane such that ...

\textit{[A diagram shows a simple truss A-C-B, with C above the line AB, and a vertical member from ...

The normals from a point to the cubic $ay^2=x^3$ make angles with the axis of $x$ whose sum is $\alp...

A triangle is circumscribed to a circle of given radius $r$, and the sides of the triangle are to be...

A triangle moves so that each of two sides passes through a fixed point. Prove that its base touches...

In a triangle prove that \[ \text{(i) } a = \frac{r_1(r_2+r_3)}{\sqrt{r_2r_3+r_3r_1+r_1r_2}}, \q...

Forces $X, Y, Z$ act along the sides $BC, CA, AB$ of a triangle $ABC$ (supposed not equilateral), an...

Assuming the earth's surface to be spherical, show that the mean distance from the north pole of all...

Small errors $\delta a, \delta b, \delta c$ are made in measuring the sides of a triangle; prove tha...

If the cross ratios of the two ranges $PQRS$ and $PXYZ$, having the point $P$ in common, are equal, ...

$A$ is a fixed point outside a given fixed circle, and $P$ is any point on the circumference. The li...

Prove that the radical axis of a fixed circle and a circle which passes through two given points pas...

Points $L, M, N$ are taken in the sides $BC, CA, AB$ of a triangle. Prove that the normals to the si...

Prove that if $\lambda, \mu, \nu$ are such that \[ \lambda(ax^2+2hxy+by^2+2x) + \mu(a'x^2+2h'xy+b'y...

Define a couple and establish the principal properties of a couple. The figure represents the horizo...

The angles of elevation of the top of a mountain from three points $A, B, C$ in a base line are obse...

In what sense is a couple a vector? Give reasons for your answer. \par If forces completely repr...

An aeroplane is travelling in a straight line with constant velocity $v$ feet per second at a consta...

Eliminate $x, y$ and $z$ from the equations \begin{align*} \frac{x}{y}+\frac{y}{z}+\frac...

From a house on one side of a street observations were made of the angle subtended by the height of ...

Solve the equations: \begin{align*} y^2+z^2-x(y+z) &= a \\ z^2+x^2-y(z+x) &= b \...

The base $a$ of a triangle and the ratio $r(<1)$ of the sides are given. Prove, geometrically or oth...

An aeroplane is travelling in a horizontal line with constant velocity. Explain how two observers at...

Shew that if four points are chosen so that two rectangular hyperbolas can be drawn to pass through ...

Two tetrahedra are such that lines joining corresponding vertices meet in a point, prove that pairs ...

A regular tetrahedron formed of light rods freely jointed to each other at their ends is suspended f...

$S$ is the area of a quadrilateral of which $a,b,c,d$ are the sides, $x,y$ the diagonals, and $2\alp...

A particle is fastened to a straight elastic string the ends of which are tied to two fixed points. ...

By drawing the graph of $y=\sin x$, prove that the equation $x=10\sin x$ has seven real roots....

A pole DE, inclined to the vertical, stands at D on a horizontal plane, and A, B, C are three collin...

Give some account of the theory of a framework of rods, dealing with (i) the number of rods necessar...

Four points $A,B,A',B'$ are given in a plane: prove that there are always two positions of a point $...

Points $X,Y,Z$ are taken on the sides $BC,CA,AB$ of a triangle $ABC$, and the circumcircle of the tr...

Solve the equations \begin{align*} (y-c)(z-b) &= a^2, \\ (z-a)(x-c) &= b^2, \\ ...

Prove that the circumcircle of the triangle formed by three tangents to a parabola passes through th...

Two motor cars $A, B$ are travelling along straight roads at right angles to one another, with unifo...

(i) $p_1, p_2, p_3, p_4$ are the lengths of the perpendiculars from the vertices of a tetrahedron $A...

Shew that, if $a_1, a_2 \dots a_n$ are unequal positive numbers, then \[ \frac{a_1+a_2+\dots+a_n...

Solve the equations:- \begin{enumerate} \item[(i)] $\sqrt{x-a}+\sqrt{x-b}+\sqrt{x-c}=0$,...

If P is the orthocentre of the triangle ABC, O its circumcentre and I its incentre, prove that \...

Find the sum of $n$ terms of the series: \begin{enumerate} \item[(i)] $\sin^2\alpha + \sin^2 2...

Explain the application of Bow's notation in the graphical solution of certain statical problems. T...

A function of $x$ is defined for positive values of $x$ by the equation \[ f(x) = \int_1^x \frac{du...

Points $P, Q$ are taken in the sides $AB, CD$ respectively of a quadrilateral $ABCD$ so that $AP:PB:...

Each generator of a cylinder touches a sphere of radius $a$. Two planes are taken perpendicular to t...

Prove the formula $F = \frac{h^2}{p^3}\frac{dp}{dr}$, for a particle describing a plane orbit under ...

The tangents to the circumcircle of a triangle $ABC$ cut the opposite sides in $X, Y, Z$. Prove the ...

Two figures $ABC..., A'B'C'...$ in the same plane are related in such a way that points correspond t...

Prove that a continuous function of one variable is bounded in any interval in which it is continuou...

A point $P$ moves in a plane with a velocity compounded of two equal constant velocities, one in a f...

A soap film is attached to fixed wires in the form of one or more closed curves. Assuming that the f...

A peg is fixed in a horizontal table and a lamina with a straight slot cut in it is placed on the ta...

Explain carefully what you understand by `reversibility' as applied to a heat engine. Why, and in wh...

Reciprocate, with respect to the focus, the theorem that the circumcircle of the triangle formed by ...

$P$ and $Q$ are the intersections of the line $lx + my + n = 0$ with the parabola $y^2 = 4ax$. The c...

A room has a square horizontal ceiling of side $a$, and vertical walls of height $h$. A spider is lo...

A hole of circular cross-section is drilled through a spherical ball of radius $a$, so that the axis...

Show that if $S=0$ and $S'=0$ represent the cartesian equations of two circles, then $S+kS'=0$ also ...

Find the relation between $p$ and $\alpha$ in order that the straight line \[ x\cos\alpha+y\sin\alph...

A point $P$ moves on the quadrant of the circle $x^2+y^2=1$ for which $x\ge0, y\ge0$. The circle wit...

If the tangents at the points $P, Q$ of a parabola meet at $T$, prove that the circle $TPQ$ passes t...

The equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0, \] referred to rectangular cartesian...

Two triangles $ABC, A'B'C'$ are related so that, with respect to a given conic $S$, the polar of $A$...

A variable circle passes through a fixed point $A$ and cuts at right angles a given circle whose cen...

Find the coordinates of the centres of circles which pass through the point $(1, 1)$ and touch the a...

A chord of a circle cuts two fixed parallel chords so that the rectangles contained by the segments ...

(i) $AOA', BOB'$ are two chords of a conic, and $P, Q$ are two points on a line through $O$. Shew th...

Prove that the circumcentre $O$, the centroid $G$, and the orthocentre $H$, of a triangle $ABC$ are ...

Given two circles (the centre of each of which lies inside the other), show how to draw a rhombus $A...

Consider some of the chief results and formulae of analytical geometry in rectangular cartesian coor...

Find the equation of the circle which passes through the origin, has its centre on the line $x+y=0$,...

Prove that the middle points of a system of parallel chords of the curve \[ ax^2+2hxy+by^2=1 \] ...

An equilateral triangle has its centre at the origin and one of its sides is $x+y=1$, find the equat...

Two opposite sides of a quadrilateral inscribable in a circle lie respectively along the coordinate ...

$O$ is the middle point of a straight line $AB$ of length $2a$. $P$ moves so that $AP.BP = c^2$. She...

Show how to perform any \textbf{three} of the following constructions, using a ruler only. Justify y...

Prove that the locus of the centre of a circle, which touches two given circles in a plane, consists...

Given two intersecting straight lines and a point in a plane, shew how to draw the straight line, th...

(i) Prove that one diagonal of the parallelogram formed by the lines \[ ax+by+c=0, \quad ax+by+c...

From a variable point $P$ of the line $p \equiv ax+by+c=0$ a perpendicular $PL$ is drawn to the line...

Show that the circles with respect to which a fixed line \[ ax+by+c=0 \] is the pola...

The coordinates of the vertices of a triangle referred to rectangular axes are $(R \cos\alpha, R\sin...

If A, B, C, D are four coplanar points, prove that the three pairs of lines through any point P para...

The equation of the pair of lines $OA, OB$ referred to rectangular Cartesian axes is $ax^2+2hxy+by^2...

The homogeneous coordinates $(x, y, z)$ of a point are so chosen that the equation of the line at in...

$AOB, COD$ are two chords of a circle; shew that the triangles $AOD, COB$ are similar and hence that...

Find the condition that, if two straight lines are represented by the general equation of the second...

Shew that, if two pencils of four rays have the same cross ratio and one ray in common, then the int...

If $m_1, m_2, m_3$ are three points of a circle $C$ of radius $R$, find the limiting value of the ra...

Eliminate $x$ and $y$ from the equations \[ ax^2+by^2=1, \quad a'x^2+b'y^2=1, \quad lx+my=1, \] ...

Find the condition of perpendicularity of two straight lines whose equations are given in trilinear ...

A circle passes through a fixed point and determines an involution on a fixed straight line. Prove t...

Prove that the two straight lines \[ x^2 \sin^2\alpha \cos^2\theta + 4xy \sin\alpha \sin\theta + y^2...

$AB$ is a diameter and $P$ any point of a circle $S$. The tangent to $S$ at $P$ meets $AB$ produced ...

Show that $fyz+gzx+hxy=0$ is the equation in homogeneous coordinates of a conic circumscribing the t...

Two of the normals from a point $P$ to a given parabola make equal angles with a given straight line...

Prove that the two circles \[ (x-\alpha)^2+(y-\beta)^2 = \lambda(x^2+y^2), \quad (\alpha+\mu\bet...

Prove that the equations of two circles cutting at right angles may be put in the form \[ x^2+y^...

Two circles are given. Show how to construct a rhombus $ABCD$ with $A, C$ on one circle and $B, D$ o...

Find the angle between the lines given by the equation \[ ax^2+2hxy+by^2=0, \] and obtain the eq...

Define the radical axis of two circles and prove that the difference of the squares of the tangents ...

Four straight lines in a plane are drawn so that $AB, CD$ intersect in $E$, and $AD, BC$ intersect i...

Prove that any straight line is cut in pairs of points in involution by conics passing through four ...

Given two circles and a point $A$ on one of them, shew how to draw a chord $BA$ of one circle such t...

Prove that for different values of $p$ the centroid of the triangle whose sides are \[ x\cos\alp...

Find the equation of the straight lines that bisect the angles between the straight lines \[ ax^...

Shew that if $AOA', BOB', COC'$ are chords of a conic, and $P$ is any point on the conic, then the p...

Any chord of a circle passes through a fixed point $O$. Prove that the tangents at the ends of the c...

Prove that the sum of the squares on the four sides of a quadrilateral is equal to the sum of the sq...

Prove that if two circles cut orthogonally, any line through the centre of either is divided harmoni...

Find the condition that the pair of straight lines represented by the equation \[ ax^2+2hxy+by^2...

Chords are drawn from the origin to the parabola $2y = ax^2+2bx+c$; prove that their middle points l...

Find the equation of the bisectors of the angles between the straight lines \[ Ax^2+2Hxy+By^2=0. \] ...

Prove that the equation \[ y=x+\cfrac{c^2}{x+\cfrac{c^2}{x+\dots}} \text{ to infinity} \] re...

The origin of a pair of rectangular axes in a plane is transferred to the point $a, b$, and the axes...

Find the length of the perpendicular from the points $(h,k)$ on the straight line $x\cos\alpha+y\sin...

The diagonals of a parallelogram are the straight lines whose equation referred to rectangular coord...

Determine the condition that the general equation of the second degree \[ ax^2+2hxy+by^2+2gx+2fy...

Prove that the equation of the straight lines, which bisect the angles between the straight lines wh...

The sides of a parallelogram are $a$ and $b$ and the acute angle between them is $\alpha$; the acute...

Prove that the equations of any two circles can be put in the form \[ x^2+y^2+2kx+c=0 \quad \tex...

Prove that if the ends of each of two diagonals of a complete quadrilateral are conjugate points wit...

Obtain the formulae of transformation from trilinear co-ordinates $\alpha,\beta,\gamma$ referred to ...

Find the equation of the axes of the conic $ax^2+2hxy+by^2+2gx+2fy+c=0$. Shew that if $\displays...

A variable point $X$ is taken on the side $BC$ of a quadrilateral $ABCD$; and the line drawn through...

If $a, b, c$ are three constants, all different, show that the system of equations \begin{align*} x+...

If the equations \[ \frac{x}{y+z}=a, \quad \frac{y}{z+x}=b, \quad \frac{z}{x+y}=c, \] have a solutio...

If $a, b, c$ are three constants, all different, show that the equations \begin{align*} ...

Simplify: \begin{enumerate} \item[*(1)] $\frac{ab(a+b)+a^3+b^3}{ab(a-b)-a^3+b^3}$. ...

Solve the equations \[ \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 1\frac{2}{3}, \quad \frac{y}{x}...

$x_1, x_2, y_1, y_2, z_1, z_2$ are given. Shew that the numbers \begin{align*} X &= \lambda x_...

Solve the equations: \begin{enumerate} \item[(i)] $\sqrt{x^2+12y} + \sqrt{y^2+12x} = 33, \quad x...

Solve the equations: \begin{enumerate} \item[(i)] $\frac{x^3}{3} + \frac{y^3}{5} = 9, \q...

Solve the equations \begin{enumerate} \item[(i)] $x(y+z) = y(z+x) = z(x+y) = a^2$, ...

Solve the equation $(x+b+c)(x+c+a)(x+a+b)+abc=0$. Eliminate $x, y$ from $x+y=a, x^3+y^3=b^3, x^5+y...

If the equations \begin{align*} axy+bx+cy+d&=0, \\ ayz+by+cz+d&=0, \\ azw+bz+cw+d&=0, \\ awx+bw+cx+d...

Solve the equations: \begin{enumerate} \item[(i)] $(x-3)^{\frac{1}{2}} + (x-6)^{\frac{1}...

Solve the equations: \begin{enumerate} \item[(i)] $x+y=(1+xy)\sin\alpha, \quad x-y=(1-xy...

Solve the system of equations: \begin{align*} yz+by+cz &= a^2-bc, \\ zx+cz+ax &= b^2-ca, \...

Solve the equations \[ x^2+y^2-3x+1=0, \quad 3y^2-xy+2x-2y-3=0. \]...

Show that if $n$ straight lines are drawn in a plane in such a way that no two are parallel and no t...

\begin{enumerate}[label=(\alph*)] \item Imagine that you are writing down integers in increasing ord...

The distant island of Amphibia is populated by speaking frogs and toads. They spend much of their ti...

Let $N = p_1^{a_1} \cdots p_r^{a_r}$, where $p_1, \ldots, p_r$ are distinct primes and $a_1, \ldots,...

Given two sets $A$ and $B$, we define the symmetric difference \[A\triangle B = (A \cap B^c) \cup (A...

An harmonious population with ample space and food is liable to grow at a rate proportional to its s...

\begin{enumerate} \item[(i)] Prove that $n^5 - n$ is divisible by 30 for every integer $n$. ...

A magic square of order $n \geq 3$ is an arrangement of the numbers 1 to $n^2$ in a square so that t...

An even integer $2n$ is said to be $k$-powerful if the set $\{1, 2, \ldots, 2n\}$ can be partitioned...

$p, n$ are positive integers with $p$ a prime ($\geq 2$). Prove that the highest power of $p$ that d...

Let $N$ be the set of positive integers and $f$ a function from $N$ to $N$. Define, for $k \in N$ an...

In a class of students, feelings are running high. Those who are not friends are enemies. Every two ...

A set $S$ of positive integers is called sparse if the equation $x - y = z - t$ has no solutions wit...

Prove that by the end of a party, attended by $n \geq 2$ people, there are two people who have made ...

Show that, if $n > 1$, $1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}$ is not an integer. \te...

(i) Prove that, if $a$, $b$, $c$ are in arithmetical progression, so are $$b^2 + bc + c^2, \quad c^2...

Prove that, if $x$ is any positive integer, then $x^5 - x$ is divisible by 30. Deduce, or prove othe...

If $n$ is a positive integer and $p$ a prime number, $\alpha_p(n)$ denotes the greatest integer $k$ ...

A set of points $S$ in the plane is called \emph{convex} if, for every pair of points $P$, $Q$ in $S...

When $x$ is a real number, the notation $[x]$ (the 'integral part' of $x$) is used to denote the gre...

Let $a_1, \ldots, a_n$ be $n$ real numbers such that $0 > a_i \geq -1$ for each $i$. Prove that $$(1...

Show that if an integer of the form $4n + 3$ is expressed as a product of integers, then one at leas...

In a certain examination the possible marks were integers from 0 to 100; for each such integer there...

The sum $s(m,n)$ is defined by \[ s(m,n) = \sum_{r=1}^n \frac{1}{r}, \] where $n \geq m \geq 2$. Sho...

Show that with $n$ rods of lengths $1, 2, 3, \ldots, n$ it is possible to form exactly $\frac{1}{24}...

Show that a plane is divided by $n$ straight lines, of which no two are parallel and no three meet i...

A certain odd integer $n$ is expressed as a sum of two squares in two different ways, \[ n = x^2...

(i) Prove that $n^7-n$ is divisible by 42 for every positive integer $n$. (ii) Prove that a number o...

Let $q_n$ ($n=1,2,\dots,N$) be a set of positive numbers, not necessarily in ascending order of magn...

Show that if $a, b, c$ are real numbers different from $\pm 1$ and such that \[ a^2+b^2+c^2+2abc=1, ...

A region $R$ in a Euclidean plane is said to be convex if, for each pair of points $A, B$ both lying...

Let $x$ be a real number and let $f(x)$ denote the fractional part of $x$, that is $x-[x]$, where $[...

Establish the harmonic property of the complete quadrangle. Prove that if a conic touches the three ...

Prove that, if $x>0$ and $1>p>0$, then \[ x^p - 1 \le p(x-1). \] By means of the...

Prove the theorem of Pappus that, if $A_1, A_2, A_3$ are three points on a straight line and $B_1, B...

If $p, q$ and $x$ are integers, and $4q-p^2$ is a perfect square, prove that $p$ is even and that $y...

Three circles $S_1, S_2$ and $S_3$ have a common point of intersection $O$. The remaining points of ...

Prove that, if $\alpha, \beta, \gamma$ are the distances of the corners of an equilateral triangle o...

$ABB'$ is a straight line and $CB=CB'$. Shew that the distance between the centres of the circles in...

Prove that in any triangle $ABC$ \[ \cos 2A + \cos 2B + \cos 2C = -1 - 4 \cos A \cos B \cos C, \] ...

Prove that the line joining the circumcentre and the orthocentre of the triangle $ABC$ makes with $B...

Prove that, if $\alpha + \beta + \gamma = 360^\circ$, \[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma -...

Shew that the lengths of the three shortest lines that bisect the area $\Delta$ of a triangle $ABC$ ...

Three circles $OBC, OCA, OAB$ are cut by a fourth circle through $O$ in points $P, Q, R$ respectivel...

Prove the formulae \begin{enumerate} \item[(i)] $\Delta = \frac{s^2}{\cot\frac{1}{2}A + ...

Prove that the inscribed circle of the triangle $ABC$ will pass through the centre of perpendiculars...

Given two tetrahedra $ABCD, A'B'C'D'$, such that the lines $AA', BB', CC', DD'$ are concurrent at $O...

$P, Q, R$ are points on the sides $BC, CA, AB$ of a triangle $ABC$, and are not collinear. $QR$ meet...

Three forces $P, Q, R$ act along the sides $BC, CA, AB$ of a triangle $ABC$, and are in equilibrium ...

Explain and illustrate the principle of duality in projective geometry, and discuss the bearing on t...

The sides of a triangle are 207, 480, 417; prove that one angle is 60° and find the others....

The side $BC$ of a triangle $ABC$ is divided at $D$ so that $BD:DC=m:n$, where $m+n=1$. Prove that, ...

Find an equation connecting the expressions \[ \cos A + \cos B + \cos C, \] \[ \sin A \sin B \sin C,...

$A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ are two quadrangles such that the lines $A_1B_1, C_1D_1, A_2B_2, C...

Prove that if two coplanar triangles are such that the lines joining corresponding vertices are conc...

Prove that the internal bisector of an angle of a triangle divides the opposite side in the ratio of...

Three parallel chords of a circle, AL, BM, CN are drawn. Shew that the perpendiculars from L on BC, ...

If the feet of the perpendiculars from a point $P$ on the sides of a triangle $ABC$ are collinear, s...

Prove that the integral part of $(\sqrt{3}+1)^{2n+1}$ is $(\sqrt{3}+1)^{2n+1} - (\sqrt{3}-1)^{2n+1}$...

Explain and justify the use of the Polygon of Forces. Forces act in order along the sides of a regul...

(i) If $\alpha+\beta+\gamma = \frac{1}{2}\pi$, prove that \[ (\sin\alpha+\cos\alpha)(\sin\beta+\...

In a triangle prove that \begin{enumerate} \item[(i)] $r_1+r_2+r_3-r=4R$; \item[...

Define the nine points circle of a triangle and establish the property from which it takes its name....

The points A, B, C lie on a straight line, and P is a point not on the line. The centres of the circ...

Eliminate $x, y$ from the equations \[ \tan x + \tan y = a, \quad \sec x + \sec y = b, \quad \si...

Prove that, if x and y are unequal, and \[ x(1-yz) = (x^2-1)(y+z), \quad y(1-zx)=(y^2-1)(z+x), \...

Two triangles $ABC, A'B'C'$ are such that lines through $A,B,C$ parallel respectively to $B'C', C'A'...

Prove that, if $a, b, c$ are different positive quantities, \[ a^3+b^3+c^3 > abc(a+b+c), \] ...

If $a_1$ and $a_2$ are positive numbers and if $p$ is a positive integer, shew that \[ 2^p(a_1^p...

In the triangle $ABC$, $A=60^\circ$, $b-c=4$, and the perpendicular distance of $A$ from $BC$ is 11....

ABCD is a parallelogram and E is any point in the diagonal BD. DF drawn parallel to AE meets AC in F...

Determine the length of the perpendicular let fall from any point $(h,k)$ on the line $ax+by+c=0$. P...

A line $l$ is drawn through $O$, the orthocentre of a triangle $ABC$ and meets $BC, CA, AB$ in $D, E...

Define a range of points in involution; and prove that if $\{AA', BB', CC', \dots\}$ be such a range...

Prove that if $\cos 2A + \cos 2B + \cos 2C + 4\cos A \cos B \cos C + 1 = 0$ then $A \pm B \pm C$ mus...

Prove Wilson's theorem that if $n$ is a prime number $1+(n-1)!$ is divisible by $n$. If $n$ and $n...

Prove that the number of primes is infinite. Find $n$ consecutive numbers, none of which are primes....

Prove that if the equations \[ cy^2-2fyz+bz^2=0, \quad az^2-2gzx+cx^2=0, \quad bx^2-2hxy+ay^2=0 \] a...

If $\alpha+\beta+\gamma=2m\pi$, where $m$ is an integer, prove that \[ \cos^2\alpha+\cos^2\beta+\cos...

Prove that the circles described on the three diagonals of a complete quadrilateral are coaxal....

A conic, inscribed in a triangle ABC, touches BC, CA, AB, at A', B', C', respectively. Shew that if ...

In the case of a triangle with the usual notation, prove that \begin{enumerate} \item[(i...

Prove Pascal's Theorem that the three points of intersection of the opposite sides of a hexagon insc...

$O$ is the circumcentre, $G$ the centroid and $H$ the orthocentre of a triangle. Prove that $O, G$ a...

Prove that the volume of a parallelepiped constructed by drawing through the opposite edges of a tet...

Show that if the lines joining the points $X, Y$ on the respective sides $AB, AC$ to the opposite co...

If $\alpha+\beta+\gamma+\delta=2\pi$, show that \[ (\sin 2\alpha+\sin 2\beta+\sin 2\gamma+\sin 2...

Prove that in any triangle, with the usual notation, \begin{enumerate} \item[(1)] $4R\Delta = ...

If $n$ is a prime number, prove that $n-1+1$ is divisible by $n$. Prove that the number formed b...

Prove the formula \[ \cos(A+B) = \cos A \cos B - \sin A \sin B \] for all real values of $A$...

Prove that \[ \sin(A+B) = \sin A \cos B + \cos A \sin B, \] taking $A, B, A+B$ to be acute angles....

If the median from the vertex $B$ of an acute-angled triangle $ABC$ makes an angle $\alpha$ with $BA...

In any triangle prove the formulae \begin{enumerate} \item[(i)] $\sin\frac{A}{2} = \sqrt...

Prove that if two planes are each perpendicular to a third plane, their line of intersection is perp...

$ABC$ and $A'B'C'$ are two triangles such that $AA', BB'$ and $CC'$ meet in a point. Prove that the ...

Prove that the number of prime numbers is infinite. Prove that $(2n+1)^5-2n-1$ is divisible by 2...

In a triangle $ABC$ prove that $\frac{r}{R} = 4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}$. P...

Prove that \begin{enumerate} \item[(i)] $\cos36^\circ-\cos72^\circ=\frac{1}{2}$. ...

The bisector of the angle A of a triangle ABC meets the circumcircle in D. Prove that the line joini...

If two triangles have the three sides of the one equal to the three sides of the other each to each,...

Prove the geometrical proposition corresponding to the algebraic identity \[ a^2-b^2=(a+b)(a-b)....

Define a tangent to a circle. Prove that the tangent at any point of a circle and the radius thr...

If three concurrent straight lines drawn from the angular points $A, B, C$ of a triangle cut the opp...

Prove the formulae in the case of a triangle: \begin{enumerate} \item[(i)] $r=4R\sin\fra...

Prove that if the lines joining corresponding vertices of two coplanar triangles are concurrent, the...

Prove that, in a triangle $ABC$, if $x,y,z$ are the lengths of the perpendiculars from $A,B,C$ on th...

Four coplanar lines, taken in sets of 3, form 4 triangles; prove that the circumcircles of these 4 t...

Prove that, if $A+B+C+D=\pi$, \[ \cos 2A+\cos 2B - \cos 2C - \cos 2D = 4(\cos A\cos B\sin C\sin ...

The lengths of the perpendiculars from the angular points of a triangle on the straight line joining...

Prove that the geometric mean of $n$ positive numbers cannot exceed their arithmetic mean. Deduce th...

\begin{enumerate}[label=(\roman*)] \item Show that $8(p^4 + q^4) > (p + q)^4$. \item If $a > b > c$ ...

\begin{enumerate}[label=(\alph*)] \item Let $a, b, c$ be real numbers with $a > 0$. Prove that $ax^2...

State an inequality between the arithmetic mean of $k$ positive numbers and their geometric mean. Th...

If $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ are real numbers prove, by considering the minimum valu...

(i) Prove that \[\left(\sum_{i=1}^n a_ib_i\right)^2 \leq \sum_{i=1}^n a_i^2 \sum_{i=1}^n b_i^2,\] de...

The positive numbers $p$ and $q$ are such that $\frac{1}{p} + \frac{1}{q} = 1$. Prove that $$ab = \f...

Prove that, if $a, b$ are real, \[ ab \le \left(\frac{a+b}{2}\right)^2, \] and deduce that, if $a, b...

The numbers $a_1, a_2, \dots, a_n$ are positive and not all equal, and their arithmetic and geometri...

If $a_1, a_2, \dots, a_n$ are all positive, and $s_r = a_1^r + a_2^r + \dots + a_n^r$, prove that $n...

Prove the following inequalities: \begin{enumerate}[(i)] \item $3(x^3+y^3+z^3) > (x+y+z)...

If $a,b,c,x,y,z$ are all real numbers, and \[ a+b \ge c, \quad b+c \ge a, \quad c+a \ge b, \] show t...

The sides of a triangle are $a, b, c$ and the corresponding angles $A, B, C$. Prove that \begin{...

Prove that the geometric mean of a finite set of positive numbers is not greater than their arithmet...

The numbers $a_1, b_1, a_2, b_2, \dots$ and the numbers $c_1, c_2, c_3, \dots$ are all positive and ...

Prove that the geometric mean of a finite set of positive numbers cannot exceed the arithmetic mean,...

If $x_1, \dots, x_n; y_1, \dots, y_n$ are real numbers, prove that \[ (x_1^2 + \dots + x_n^2)(y_1^2 ...

If $a_1, a_2, \dots, a_n$ are all positive, prove that \[ \frac{a_1+a_2+\dots+a_n}{n} \ge (a_1a_...

Prove that, if $a_1, \dots, a_n$ are positive, \[ \frac{1}{n}(a_1+\dots+a_n) \ge (a_1a_2\dots a_n)^{...

If $x_1, \dots, x_n$ are real numbers, prove that \[ n(x_1^2+\dots+x_n^2) \ge (x_1+\dots+x_n)^2 ...

Show that the arithmetic mean $A=(a_1+\dots+a_n)/n$ of $n$ positive numbers $a_1, \dots, a_n$ is nev...

Prove that the geometric mean of a number of positive quantities can never exceed their arithmetic m...

Prove that the arithmetic mean of a set of positive numbers cannot be less than their geometric mean...

If $pu+qv+rw=1$, where $p, q, r, u, v, w$ are all positive quantities, prove that \[ \frac{p}{u} + \...

Show that if $p>q>0$ and $x$ is positive then \[ \frac{1}{p}(x^p-1) > \frac{1}{q}(x^q-1). \] ...

Prove that the arithmetic mean of a set of unequal positive quantities is greater than their geometr...

Prove that the arithmetic mean of $n$ positive numbers is greater than their geometric mean, unless ...

Establish necessary and sufficient conditions that $ax^2+2bx+c$ shall be positive for all real value...

Shew that the geometric mean of $n$ positive numbers is not greater than their arithmetic mean. ...

Show that, if $p>q>0$ and if $x>0$, then \[ \frac{x^p-1}{p} > \frac{x^q-1}{q}. \] ...

Prove that the arithmetic mean of a number of positive quantities is never less than their geometric...

The function $f(x)$ is such that \[ \frac{f(c)-f(b)}{c-b} > \frac{f(b)-f(a)}{b-a} \]...

If $\alpha$ is a fixed positive number less than unity, show that the least value of $a$ for which \...

Show that $y = \frac{(x - \alpha)(x - \beta)}{x - \gamma}$ can take all values as $x$ varies provide...

Shew that the geometric mean of $n$ positive numbers is not greater than their arithmetic mean. ...

If all the numbers $a_i, b_i$ and $c_i$ are positive, and if $m$ is a positive integer, shew that ...

\begin{enumerate}[label=(\roman*)] \item Show that the product of three consecutive positive integer...

Prove that $\sum_{r=1}^n r(r+1)(r+2)\ldots(r+s-1) = n(n+1)\ldots(n+s)/(s+1).$ Evaluate $\sum_{r=1}^n...

(i) Guess an expression for \[\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right) \cdots \left(1-\f...

\begin{enumerate}[label=(\alph*)] \item Consider the sequence $\{a_n\}$ of positive real numbers def...

Two sequences $(x_0, x_1, x_2, \ldots)$ and $(y_0, y_1, y_2, \ldots)$ of positive integers are defin...

Show, by induction or otherwise, that, if $n$ consecutive integers have arithmetic mean $m$, then th...

Let $f_N(a) = \underset{R}{\textrm{Max}}(x_1 x_2 \ldots x_N)$, where $a \geq 0$, and $R$ is the regi...

Suppose $f$ is a twice differentiable function with $f''(x) < 0$ for all $x > 0$. Show that if $0 < ...

The sequence $a_0$, $a_1$, $a_2$, $\ldots$ is defined by $$a_0 = a_1 = 1; \quad a_n = a_{n-1} + a_{n...

\begin{enumerate} \item[(i)] Prove, by induction or otherwise, that $3^{2n+1} + 2^{n+2}$ is divisibl...

Discover a general formula of which \begin{align} 1^3 + 3^3 + 5^3 &= 9 \times 17,\\ 1^3 + 3^3 + 5^3 ...

A finite sequence of real numbers $u_0$, $u_1$, $\ldots$, $u_n$ satisfies $$(u_{k+1} - 2u_k)^2 = 1 \...

If, for $n = 1, 2, 3, \ldots$, the polynomial $$\frac{1}{n!}x(x-1)(x-2)\cdots(x-n+1)$$ is denoted by...

Given that $a_r, b_r$ and $c_r$ are all real and positive numbers for $r=1, 2, \dots, n$, and that \...

If \[ p_r = \frac{1 \cdot 4 \cdot 7 \dots (3r-2)}{3 \cdot 6 \cdot 9 \dots (3r)} \quad (r=1, 2, \dots...

Show, by induction or otherwise, that \[ \tan(\theta_1+\theta_2+\dots+\theta_n) = \frac{\s...

(i) Prove that the sum of the cubes of the first $n$ integers is equal to the square of the sum of t...

Prove that if $n$ is a positive integer: \begin{enumerate} \item $\frac{1}{(2n!)^2} - \frac{1}{(...

Prove that \[ 1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{1}{6}n(n+1)(2n+1), \] and evaluate \[ 1^3 ...

Shew (by induction or otherwise) that if $n$ and $k$ are positive integers, then \[ f_{n,k} = x^...

By induction or otherwise prove that \[ \frac{1}{n+1} - \frac{m}{n+2} + \frac{mC_2}{n+3} - \frac{mC...

Explain the principle of proof by 'mathematical induction'; and prove in this way that \[ ...

Find the sum of the cubes of the first $n$ natural numbers. Find the sum to $2n+1$ terms of the ...

Prove that the geometric mean of $n$ positive numbers does not exceed their arithmetic mean. Shew th...

Prove that \begin{enumerate} \item If $n$ is an integer greater than 2, then \[ ...

If $n$ is a positive integer, prove that $3 \cdot 5^{2n+1} + 2^{3n+1}$ is divisible by 17 and $3^{2n...

Eliminate $\theta$ from the equations \[ a\sin\theta + b\cos\theta = a\operatorname{cosec}\theta...

If $n$ is a positive integer, prove that \begin{enumerate} \item[(i)] $n^5 - 4n^3 + 5n^2 - 2n$...

Find the sum of the cubes of the first $n$ natural numbers, and determine a set of $2n+1$ consecutiv...

Sum the series: \begin{enumerate} \item[(i)] $2 \cdot 2! + 3 \cdot 3! + 4 \cdot 4! + \dots$ to $...

Explain the method of proving theorems by mathematical induction. Shew that the series \[ \f...

Shew by induction or otherwise that the sum of $n$ terms of the series \[ 1 + \frac{n-1}{n-\frac...

Prove by induction that the square of the sum of the cubes of the first $n$ integers is the arithmet...

Prove that, if $a$ and $b$ are positive integers, \begin{enumerate} \item $\frac{a^5}{12...

If £$P$ is the present value of an annuity of £$A$, to continue for $n$ years, at $100r$ per cent. p...

\begin{enumerate} \item[(i)] If $y=x^{n-1}\log x$, prove that \[ x\frac{d^n y}{dx^n}=n-1...

Explain briefly the method of mathematical induction and give an illustration of its use. Prove ...

Explain the method of mathematical induction and use it to prove that if \[ {}^nS_r = 1^r+2^r+\d...

Solve the equations \begin{enumerate} \item[(i)] $x+y+z=x^2+y^2+z^2 = \frac{1}{3}(x^3+y^...

Find the sum of the squares of the first $n$ natural numbers. Find the sum of all possible produ...

Prove that the arithmetical mean of any number of positive quantities is greater than their geometri...

Show that, if $p = \cos A + \cos B$ and $q = \sin A + \sin B$, then $\sin(A + B) = \frac{2pq}{p^2+q^...

$C$ is a circle of radius $r$. Determine the length $l$ of the side of a regular $n$-sided polygon i...

The sides of a triangle are $p$, $q$, $r$; the angles opposite them are (in circular measure) $P$, $...

A spaceship is constructed by attaching the plane circular face of a hemisphere of radius $a$, to th...

Show that, if $n$ is a positive integer, the equation $$2x = (2n+1)\pi(1-\cos x),$$ (where $\cos x$ ...

Prove that $\displaystyle\frac{\cot 3x}{\cot x}$ never lies between 3 and $\frac{1}{3}$....

The sine of an acute angle is equal to $\cdot 9998$, accurately; with the aid of the four-figure tab...

Express the area of a triangle (1) symmetrically in terms of $R$ the circumradius and the angles, (2...

Find the value of $\sin\left(\cos^{-1}\frac{63}{65} + 2\tan^{-1}\frac{1}{5}\right)$. Given \...

Find the only value of $x$ which satisfies the equation \[ 3\tan^{-1}x + \tan^{-1}3x = \frac{1}{4}...

If $A+B+C=\pi$, prove that \begin{enumerate} \item[(i)] $1-\cos^2A-\cos^2B-\cos^2C-2\cos...

Prove that $\tan^{-1}a + \tan^{-1}b = \tan^{-1}\left(\frac{a+b}{1-ab}\right)$. Solve the equation ...

Prove that \[ \tan 20^\circ \tan 30^\circ \tan 40^\circ = \tan 10^\circ. \] If $A+B+C=90^\circ$,...

Find a real value of $x$ making $$f(x) = -3|x|^4 + 8|x|^3 + 6|x|^2 - 24|x| - 201$$ as large as possi...

Define the product of two real $2 \times 2$ matrices. Show that this multiplication is associative. ...

A matrix $B$ satisfies $B^2 = B$ and is known to be of the following form: \[B = \begin{pmatrix} a &...

Let $A$ be any $2 \times 2$ matrix with integer entries. The trace of $A$ is defined to be the sum o...

The numbers $a, b, c, d$ have the property that there exist $x_1, x_2$, not both zero, such that \be...

Define the inverse $A^{-1}$ and the transpose $A^T$ of an invertible $n \times n$ matrix $A$. If $B$...

Let $C$ be the set of matrices of the form \begin{equation*} \begin{pmatrix} a & b \\ -b & a \end{pm...

Let $A$, $B$ be real $2 \times 2$ matrices. Show that only one of the following assertions is always...

The elements of the $n \times n$ matrix $A = (a_{ij})$ are all equal to either 1 or $-1$. Prove or d...

Two real differentiable functions $u(x)$, $v(x)$ are said to be linearly dependent in $-1 \leq x \le...

Show that the operation of matrix multiplication on the set $M_2$ of real $2 \times 2$ matrices is a...

Define the \textit{determinant} of a $2 \times 2$ matrix $C$ with complex entries, and show that $C$...

The trace of a square matrix is defined to be the sum of its diagonal elements. If $A$ and $B$ are b...

If $a$, $b$, $c$ and $d$ are all positive, prove that there is a positive value of $t$ such that the...

Given three real non-zero numbers $a$, $b$, $h$, prove that the relations \begin{align} ax + hy &= \...

The nine numbers $a_{ij}$ ($i,j=1, 2, 3$) satisfy the equations \[ a_{i1} a_{j1} + a_{i2} a_{j2} + a...

Two lines $ABC\dots$, $A'B'C'\dots$ meet in a point $O$. Shew that forces acting along $AA'$, $BB'$,...

Prove that, if \[ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy\] is the product of two factors lin...

$A_1, A_2, \dots, A_n$ are $n$ points whose coordinates are $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_...

Let $A(\theta)$ and $B(\theta)$ denote the matrices $$\begin{pmatrix} \cos\theta & \sin\theta \\ \si...

Show that $\begin{vmatrix} 1+x_1 & x_2 & x_3 & \cdots & x_n \\ x_1 & 1+x_2 & x_3 & \cdots & x_n \\ \...

The rectangular Cartesian coordinates of $P'$ are $(x', y')$ and a mapping $\alpha$ of the plane int...

The following four functions are defined for all real $x$: (i) $\log(2e^x)$; (ii) $e^x$; (iii) $|x|$...

Two lines in a plane meet in $P$. Prove that successive reflexion in the two lines is equivalent to ...

Find, in terms of $h$, $k$, $\sin 2\theta$ and $\cos 2\theta$, the co-ordinates of the mirror-image ...

The lines $AB$ and $A'B'$ are equal in length and lie in a plane. Show that $A'B'$ can always be bro...

The points in a plane are displaced so that the point $(x,y)$ referred to rectangular coordinates ta...

The line $AB$ is equal in length to $A'B'$ and in the same plane: shew that $AB$ can always be moved...

Find the two values of $\lambda$ for which the matrix equation $\begin{pmatrix} 3 & 2 \\ 1 & 4 \end{...

A square matrix $B$ has an inverse $B^{-1}$; $B$ satisfies \[BX = \lambda X\] for some scalar $\lamb...

The three numbers $X, Y$ and $Z$ are related to the three numbers $x, y$ and $z$ by the two equation...

If $\lambda_1, \lambda_2$ are the roots of the equation in $\lambda$, \[ \begin{vmatrix} a-\lamb...

The set of numbers $x_1, x_2, \dots, x_n$ are transformed into the set of numbers $\xi_1, \xi_2, \do...

Show that the equation \[ \{a(x^2-c)-b(y^2-c)\}^2 + 4\{axy+h(y^2-c)\}\{bxy+h(x^2-c)\}=0 \] repre...

Let $$p(x) = 8x^4 - 8x^2 + 1.$$ Given that $\cos 4\theta = p(\cos \theta)$, sketch the graph of $y =...

(i) Sketch, in the same diagram, the graphs of \[y = \tan x, \quad y = \tan^{-1} x,\] where $\tan^{-...

An assembly hall has a semi-circular dais of radius $a$, set with its bounding diameter against a st...

Show that \[\cos 3\theta = 4\cos^3\theta - 3\cos\theta\] (thus expressing $\cos 3\theta$ as a cubic ...

Show that, for $0 < \lambda < 1$, the least positive root of the equation $$\sin x = \lambda x \qqua...

Prove the formulae \begin{align*} \sin\frac{y}{2} \sum_{m=0}^{N} \sin my &= \sin\frac{(N+1)y}{2}\sin...

Show that \[2\sin\frac{1}{2}x \sum_{n=1}^{N} \cos nx = \sin(N + \frac{1}{2})x - \sin\frac{1}{2}x.\] ...

Prove that the orthocentre of the triangle formed by the points $(a\cos\alpha, a\sin\alpha)$, $(a\co...

$ABC$ is the triangle formed by the tangents to the circle $x^2 + y^2 = r^2$ at the points $(r\cos\t...

A flagstaff leaning due north at an angle $\alpha$ to the vertical subtends angles $\phi_1$ and $\ph...

Obtain the general solutions of the trigonometrical equations: \begin{enumerate} \item[(i)] $\sin^{-...

Find all the real roots of the two following equations in $x$: \[\cos(x\sin x) = \frac{1}{2};\] \[\c...

A man observes that the summit of a nearby hill is in a direction $x$ radians east of north, and at ...

(i) Evaluate the sum \[ \sum_{r=1}^{n} \sin^3 rx. \] (ii) By using an algebraic relation between $\t...

(i) $A, B, C, D$ are the angles of a plane quadrilateral. Show that $$4\sin(A+B)\sin(B+D)\sin(D+A) =...

If $n$, $r$, $s$ are non-negative integers, and $k$ is a positive integer, show that \begin{align} |...

Solve the following equations: \begin{enumerate} \item[(i)] $x^2(y^2 - 1) + xy + 1 = 0$,\\ $...

Show that $$x^{2n} - 2x^n \cos n\theta + 1 = \prod_{r=0}^{n-1} \left[ x^2 - 2x \cos \left( \theta + ...

If $A + B + C = \frac{\pi}{2}$, prove that \begin{align} (\sin A + \cos A)(\sin B + \cos B)(\sin C +...

Show that for each integer $n \geq 1$ there is a polynomial $T_n(x)$ of degree $n$ such that $$T_n(\...

A piece of paper has the shape of a triangle $ABC$, where $\angle BCA = \frac{1}{5}\pi$, $\angle CAB...

Let $$f(x) = k\cos x - \cos 2x,$$ where $k$ is a constant, $k > 0$. By considering the sign of $f'(x...

Solve completely the equation \[ \sin 3x = \cos 4x. \] Hence, or otherwise, find all the roots of \[...

Prove that, in general, \[ \frac{\sin x}{\sin(x-a)\sin(x-b)} \] can be expressed in the form \[ \fra...

Three equal circular arcs, each of radius $a$ and angle $\beta (<2\pi/3)$, are joined together to fo...

The ridges of two roofs are horizontal and at right angles to each other, and the inclination of eac...

If two triangles $ABC$ and $A_1B_1C_1$ are of equal area, prove that \[ \sum_{a,b,c} a^2 \cot A_1 = ...

Prove that if \[ \sec\alpha = \sec\beta\sec\gamma + \tan\beta\tan\gamma, \] then either \[ \begin{ca...

If $\alpha, \beta, \gamma$ are three real angles, and if the equations \begin{align*} x\cos\alpha + ...

Describe, with the help of a rough sketch, the form of the curve \[ x = \cos \frac{y}{x}. \] ...

$A, B, C$ are the angles of a triangle. Prove the inequalities \[ \sin A + \sin B + \sin C \ge \sin ...

By considering an isosceles triangle with base angles $\pi/5$, or otherwise, show that \[ \cos \frac...

Prove that if, in measuring the three sides of a triangle, small errors $x, y$ and $z$ are made in t...

Discuss the maxima and minima of the function \[ \sin mx \csc x, \] where $m$ is a positive ...

Find all the real solutions of the simultaneous equations \begin{align*} \sin^{-1}\tfrac{1}{2}x + \s...

The sequence $A_0, A_1, \dots, A_n, \dots$ is defined by \[ A_0=0, \quad A_{n+1}\cos n\theta - A_n \...

Express $\cos 3\theta$ in terms of $\cos\theta$. Show that, for any real $\theta$, \[ \cos\theta - \...

Sum the series \[ \sum_{r=0}^{n-1} \sin^2(\alpha+r\beta). \] Deduce that, if $0 < \beta < \frac{\pi}...

Prove that the function \[ \frac{1-\cos x}{\sin(x-a)} \quad (0 < a < \pi), \] has infinitely many ma...

A family of curves is given by the equation $y = \cos x + \lambda \cos 3x$, where $\lambda$ is a pos...

The sides $a,b,c$ of a triangle are measured with a small percentage error $\epsilon$ and the area i...

Express $\cos 2\theta$ and $\sin 2\theta$ in terms of $\tan \frac{1}{2}\theta$. Find all values ...

Find, to the nearest minute, all angles $x$ and $y$ for which \begin{align*} \tan \tfrac...

Find the numerical values of \[ y = \sin\left(x+\frac{\pi}{4}\right) + \frac{1}{4}\sin 4...

Obtain by a geometrical construction, or otherwise, the solutions of the equations \begin{align*} ...

Eliminate $\theta$ and $\phi$ from \begin{align*} \sin\theta + \sin\phi &= a, \\ ...

Write down the most general values of $x$ which satisfy the equations \begin{enumerate} ...

Shew that if $\alpha + \beta + \gamma = \frac{\pi}{4}$, then \[ (\sin\alpha + \cos\alpha)(\sin\beta...

Prove that \[ 2 - 2 \cos \theta + \cos 2\theta - 2 \cos 3\theta + \cos 4\theta \ge 0. \] ...

A vertical tower of height $h$ stands on the top of a hill and the angles of elevation of the top of...

Prove, as simply as you can, that of the three following equations there are two which cannot be sat...

Find for what values of $k$ the equation \[ \sin x \sin 3x = k \] has real solutions in $x$....

Express $\tan 2x$ in terms of $\tan x$, and $\tan x$ in terms of $\tan 2x$. Explain the relation bet...

Prove that, if any two of \[ \sin(B+C) + \sin(C+A) + \sin(A+B) \] and the three similar function...

Solve the equations: \begin{enumerate} \item[(i)] $\cos x + \sin x = 1$, \item[(...

Solve completely the equation \[ \sin 5x = \cos 4x. \] Deduce that one root of the equation \[...

Prove that, if $x$ lies between 0° and 180°, $\cos x - \frac{1}{4} \cos 2x$ lies between $-\frac{3}{...

Prove that \[ 1-\cos^2 A - \cos^2 B - \cos^2 C + 2\cos A \cos B \cos C = 4 \sin S \sin(S-A) \sin...

(i) Find all the values of $\theta$ which satisfy the equation \[ \cos\theta + \cos 2\theta = \sin...

Prove that, if $X+Y+Z$ is equal to $2n$ right angles, where $n$ is an integer, then \[ \sin 2X + \si...

Find for what ranges of values of $\theta$ between $0$ and $\pi$ each of the following inequalities ...

Shew that, if $c^2<a^2+b^2$, $a\cos\theta+b\sin\theta+c$ has two zeros $\theta=\alpha$ and $\theta=\...

Prove that, in any triangle, $a \cot A = b \operatorname{cosec} C - a \cot C$. If $a=19.1, b=15....

Find an expression giving all the angles which have the same sine as $A$. Solve the equation \be...

Find an expression for all angles which have the same sine as $\alpha$. Find all the solutions of ...

(i) Find all the real roots of the equation \[ \tan^2 x + \tan^2 2x = 10. \] (ii) Eliminate $\th...

Show that, if $\alpha+\beta+\gamma = \pi$, \[ (\sin 2\alpha + \sin 2\beta + \sin 2\gamma)(\cot\a...

Prove that (i) if $\theta+\phi+\psi = \pi/2$, \[ \sin^2\theta + \sin^2\phi + \sin^2\psi + 2\sin\...

Prove that \[ 1+\sec 20^\circ = \cot 30^\circ \cot 40^\circ \] and solve the equation \[...

Simplify the fraction $(cos 3\theta + \cos 4\theta)/(2\cos 3\theta - 2\cos 2\theta + 2\cos\theta - 1...

\begin{enumerate} \item[(i)] Solve the equations \begin{align*} \tan x + \tan y &= 4, ...

If \[ (1+3\sin^2\phi)^{\frac{1}{3}} = \sin^{\frac{2}{3}}\theta + \cos^{\frac{2}{3}}\theta, \] ...

Prove that \[ 2\tan^{-1}\left(\tan\frac{\theta}{2}\tan\frac{\phi}{2}\right) = \cos^{-1}\left(\fr...

Given the sides of a triangle, find an expression for the tangent of half of one of its angles. ...

If $\tan 4\theta = \tan 4\alpha$, express in terms of the trigonometrical ratios of $\alpha$ the pos...

Solve the equation \[ 2\sin x.\sin 3x=1. \] If \[ \tan\beta = \frac{n\sin\alpha.\cos\alpha}{1-n\sin^...

Prove that \[ 1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha.\cos\beta.\cos\gamma = 4\sin s...

If \[ \tan\phi = \frac{\sin\alpha\sin\theta}{\cos\theta-\cos\alpha}, \] prove that \[ \t...

Find an expression for all the values of $\theta$ satisfying the equation $\sin\theta=\sin\alpha$. ...

Prove that in a triangle $\tan\frac{A-B}{2} = \frac{a-b}{a+b}\cot\frac{C}{2}$. In a triangle $a=...

If $A+B+C=\pi$, prove that \begin{enumerate} \item[(i)] $\sin 2nA + \sin 2nB + \sin 2nC ...

Shew that $\cos\frac{A}{2} = \pm\frac{1}{2}\sqrt{1+\sin A} \pm \frac{1}{2}\sqrt{1-\sin A}$, and dete...

Draw the graphs of $\cot x$ and $e^x\sin x$. Find the tangents of the angles which satisfy the e...

Prove geometrically that $\tan(A+B)(1-\tan A\tan B) = \tan A+\tan B$, assuming that $A+B<\pi/2$. ...

With the usual notation prove that \[ r = 4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}. \] ...

Prove geometrically \begin{enumerate} \item[(i)] $\tan\frac{A}{2} = \frac{\sin A}{1+\cos...

Prove \begin{enumerate} \item[(i)] $\tan 6^\circ \cot 12^\circ \cot 24^\circ \cot 48^\ci...

Prove geometrically that $\tan\theta = \csc 2\theta - \cot 2\theta$. If $\alpha+\beta+\gamma = \...

Prove that \[ \sin A + \sin B + \sin C - \sin(A+B+C) = 4\sin\frac{1}{2}(B+C)\sin\frac{1}{2}(C+A)\s...

Eliminate $\theta, \phi$ from the equations: \[ a\sec\theta+b\text{cosec }\theta=c, \quad a\sec\...

Prove that if $A$ and $B$ are acute angles while $A+B$ is obtuse, \[ \cos(A+B) = \cos A\cos B - ...

Prove that \[ \sin A+\sin B+\sin C - \sin(A+B+C) = 4\sin\tfrac{1}{2}(B+C)\sin\tfrac{1}{2}(C+A)\sin\t...

Prove geometrically that $\tan A = \frac{\sin 2A}{1+\cos 2A}$. If $ABC$ is a triangle, prove that ...

Prove that $\Sigma[\cos 2A-\cos(B+C)](\cos B-\cos C) = \Sigma\sin(C-B)(\sin B+\sin C)$....

The triangle $ABC$ is inscribed in a circle $K$ of radius $R$, and its angles are all acute. If smal...

A ship is steaming due east at a constant speed. The ship sends out an SOS call which is received by...

The area of a triangle is to be determined by the measurement of its sides. If the maximum small per...

Prove that $\cot \theta - 2 \cot 2\theta = \tan \theta$. Hence or otherwise prove that: \[\frac{1}{2...

Show that the increment in the radius $R$ of the circumcircle of a triangle $ABC$ due to small incre...

The sides $a$, $b$, $c$ of a triangle are measured with a possible small percentage error $\epsilon$...

Define exactly what is meant by the derivative $dy/dx$ of a function $y = f(x)$. Obtain from first p...

An inaccessible vertical tower $CD$ of height $h$ is observed from two points $A$ and $B$ which lie ...

Prove that \[ 2^{-n}\sin\theta\operatorname{cosec}(\theta/2^n) = \cos(\theta/2)\cos(\theta/2^2)\...

Prove that the increment in the angle $A$ of a triangle due to small increments in the sides is give...

Prove that, if $0 < x < 1$, then \[ \pi < \frac{\sin\pi x}{x(1-x)} < 4. \] Sketch the graph of the f...

In a triangle $ABC$ the side $a$ and the angles $B, C$ (measured in radians) are taken as independen...

Shew that the error in taking $\frac{3\sin\theta}{2+\cos\theta}$ for $\theta$ is less than two-third...

Shew that the area of a segment of a circle of radius $r$ cut off by a chord of length $2c$, where $...

Prove that $\frac{\sin\theta}{\theta}$ diminishes steadily from 1 to $\frac{2}{\pi}$ as $\theta$ inc...

Give without proof expressions for $\sin\theta, \cos\theta$ in terms of $t \left(=\tan\frac{\theta}{...

Prove that the length of the line joining the orthocentre of a triangle $ABC$ to the middle point of...

Explain the relation between the greatest and least values taken by a function in an interval, the m...

Show that \[ (-1)^n e^{z^2} \frac{d^n e^{-z^2}}{dz^n} \] is a polynomial of degree $n$ in $z$. Call ...

Prove that the only positive integers $x$ and $y$ satisfying the conditions $x < y$ and $x^y = y^x$ ...

If \[ f(\theta) = \sum_{r=1}^n a_r \sin (2r-1)\theta, \] where $a_1 > a_2 > \dots > a_n > 0$, show b...

State and prove Leibniz' theorem concerning the $n$th derivative of a product $u(x)v(x)$. If $y=y_n(...

If the angles $\theta_1, \theta_2, \dots, \theta_n$ all lie between $0$ and $\frac{1}{2}\pi$, and $\...

Show that the stationary values of the function \[ (a-\cos t)^2 + t^2 + (b-\sin t)^2 \] are given by...

The area $\Delta$ of a triangle is expressed as a function of its sides $a,b,c$. Show that \[ \Delta...

Differentiate the following expressions: \begin{enumerate}[(i)] \item $\cos\log x$; ...

Explain how a knowledge of the solutions of the equation $f'(x)=0$ may give information about the ro...

The curve $y=ax+bx^3$ passes through the points $(-0.2, 0.0167)$ and $(0.25, 0.026)$. Prove that the...

Find any maxima, minima and points of inflexion on the curve \[ y = \frac{|x-1|}{x^2+1}. \]...

Find the maxima, minima and points of inflexion of the curve $y = \sqrt{x} \cos \log \sqrt{x}$, wher...

If $y = e^{\frac{1}{2}x^2+bx^2}$ and $c_n = \left(\frac{d^n y}{dx^n}\right)_{x=0}$, show that $c_{n+...

Prove that the $n$th derivative of \[ \frac{1}{x^2+b^2} \quad (b \ne 0)\] is \[ \frac{(-)^n n!}{b^{n...

Prove Leibniz' theorem for the $n$th differential coefficient of the product of two functions. By us...

Show that the function \[ \frac{x^3+x^2+1}{x^2-1} \] of the real variable $x$ has only two critical ...

Prove that for an algebraic equation $f(x)=0$, there can at most be only one real root in a range of...

Establish Leibniz' theorem for the $n$th derivative of the product of two functions. If $f=(px+q)/(x...

If \[ X_n = e^{-x^2} \frac{d^n}{dx^n}(e^{x^2}), \quad (n=0, 1, 2, 3\dots) \] establish the r...

If \[ L_n(x) = e^x \frac{d^n}{dx^n}(x^n e^{-x}) \] show that: \begin{enumerate} \item[(i)] $L_{n+1}(...

A man can walk at the rate of 100 yd. a minute, which is $n$ times faster than he can swim. He stand...

(i) Show that \[ |(x+y)e^{-2(x^2-xy+y^2)}| \le e^{-1/2} \] for all real $x, y$. When is the sign of ...

If $f(x)$ is a polynomial in $x$ of degree 2, and \[ F_n(x) = \frac{d^n}{dx^n} [\{f(x)\}^n], \] show...

If $f(x)$ is a polynomial and $f'(x)$ its derivative, state, without proof, what you can deduce abou...

Show that, for all real values of $x$ and $\theta$, the value of the expression \[ \frac{x^2+x \...

Prove that the radius of curvature at any point of a curve $y = f(x)$ is \[ \frac{\left\{1 + \le...

Find the $n$th differential coefficients of \[ \cos x, \quad \cos^2 x, \quad \log(1+x), \quad \f...

Differentiate \textit{ab initio} $\log x$, $\tan^{-1} x$. Differentiate $e^{\sin(\log x)}$....

Defining the curvature of a plane curve at any point as the limit of $\delta\psi/\delta s$ when $\de...

Differentiate $x^{\log x}$, $(\log x)^x$. \par Find the $n$th differential coefficient of $a^x \...

Explain what is meant by the limit of $\frac{f(x+h)-f(x)}{h}$ as $h$ converges to zero, and illustra...

Prove that \[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{1}{v^2}\left(v\frac{du}{dx} - u\frac{dv}...

Find the values of $x$ which give maxima and minima of \[ \sin x + \frac{1}{3}\sin 3x + \frac{1}...

Find the values of $x$ for which the function $e^{mx} \cos 3x$, where $m$ may be positive or negativ...

\begin{enumerate} \item[(i)] If $y\sqrt{1-x^2} = \cos^{-1}x$, prove that \[ (1-x^2)\frac{dy}...

Define the differential coefficient of a function. Has $x\sin\dfrac{1}{x}$ a differential coefficien...

Find from the definition the differential coefficient of $\sin x$, establishing the limiting value r...

Give an account of the application of the calculus to the discovery of, and the discrimination betwe...

Differentiate (i) $\dfrac{(1+x^2)^{\frac{1}{2}}+(1-x^2)^{\frac{1}{2}}}{(1+x^2)^{\frac{1}{2}}-(1-...

Differentiate with regard to $x$ \[ 2\sqrt{3}\tan^{-1}(2x+1)/\sqrt{3} - 3x/(x^3-1) - \log(x-1)^2...

Determine the stationary values of the function $e^{ax}\sin bx$, where $a$ and $b$ are positive, and...

Find from the definition the differential coefficients of \begin{enumerate} \item[(1)] $...

Trace the curve \[ x(y^2-\frac{1}{2}a^2) - y(x^2-\frac{1}{2}a^2) = a^3, \] and shew that the rad...

Find $\dfrac{dy}{dx}$ in the case where (i) $y = \sin^{-1}\left(\dfrac{b+a\cos x}{a+b\cos x}\rig...

Show that the function $\frac{\sin^2 x}{\sin(x+a)\sin(x+b)}$ ($0 < a < b < \pi$) has an infinity of ...

(a) Differentiate with respect to $x$: (i) $x^{x^{\cosh^{-1}x}}$; (ii) $\tan^{-1}\left[\tan x \fra...

Differentiate $x^{x^2}$, $(ax^2+b)^n$, $x^2 \sin x$ and $\frac{x+2}{(x+1)(x+3)}$....

Find from first principles the differential coefficients of $x^n$ and $\cos^{-1}x$. Find the $n$...

If $y = \sqrt{1-x^2}.\sin^{-1}x$, prove that \begin{enumerate} \item $(1-x^2)\frac{d^2y}{dx^2}...

Explain in what sense the Kelvin scale of temperature is ``absolute.'' How is it possible to test th...

Differentiate $\sin^{-1}\frac{a+b\cos x}{b+a\cos x}$. If $\log x + \log y = \frac{x}{y}$, prove th...

If $y=\sin(a\sin^{-1}x)$, prove that $(1-x^2)\frac{d^2y}{dx^2}-x\frac{dy}{dx}+a^2y=0$. Hence or ot...

Show that if $e(x)$ is a differentiable function with $e'(x) = e(x)$ and $e(0) = 1$ then, if $a$ is ...

A chemist wishes to conduct an experiment in which a process takes place for a fixed interval of tim...

Suppose that the functions $f(x)$ and $g(x)$ can each be differentiated $n$ times. Prove that one ca...

Prove Leibniz's theorem on the differentiation of the product of two functions. By considering $$\fr...

Suppose that the function $f(x)$ has derivatives of all orders. Show by induction that \[ \frac{d^n}...

If $y_m(x)$ is defined as a function of $x$ by the equation $$y_m(x) = (-1)^m e^{x^2} \frac{d^m}{dx^...

Explain the principle of mathematical induction, and use it to prove that the $n$th derivative of th...

Establish Leibnitz' theorem for the $n$th derivative of a product of two functions. If \[f(x) = \fra...

The functions $f_n(x)$ are defined thus: \begin{align} f_0(x) = 1, \quad f_n(x) = (-\frac{1}{2})^n e...

The functions $u(x)$ and $v(x)$ satisfy the equations \begin{align} u'' + u &= 0, & u(0) &= 0, & u'(...

If $y=\sin^{-1}x$, show that $y''(1-x^2)=xy'$. By Leibniz' Theorem or otherwise, find $y^{(n)}$ at $...

Given that $a$ and $b$ are positive constants and $x$ is a real variable, prove that \[f(x) = a \cot...

Prove that, if $f(x) = e^{ax} \sin bx$, then \[ f'(x) = r e^{ax} \sin (bx + \phi), \] and specify th...

Having given \begin{align*} ax + by &= 1, \\ a'x + b'y &= 1, \\ ab &= a'b', \\ a +...

If in a triangle $ABC$ the side $a$ is increased by a small quantity $x$ while the other two sides a...

Show that for two values of $\lambda$ the equations \begin{align*} (2+\lambda)x + 4y + 3...

Prove that \[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}. \]...

Find the $n$th differential coefficients of $x^n e^{ax}$ and $e^{ax}\sin x$, and shew that the $n$th...

Find from the definition the derivative of $\sin^{-1}x$. \par Prove that for the value $x=0$, $\...

Prove that \[ \left(\frac{d}{dx}-\tan x\right)^n u_n = n! u_0, \] where \[ u_n = x^n \sec x. \]...

Find $\frac{dy}{dx}$ in the following cases: \begin{enumerate} \item $y = \tan^{-1}x + \tan^{-...

Prove that the length of the subnormal of the curve \[ y=f(x) \text{ is } y\frac{dy}{dx}. \] ...

Prove from first principles that, if $f(x)$ is continuous in $a \le x \le b$ and differentiable in $...

Show that $y = \sin x \tan x - 2 \log \sec x$ increases steadily as $x$ increases from $0$ to $\frac...

By means of the calculus or otherwise, prove that if $p > q > 0$ and $x > 0$, then \[q(x^p - 1) > p(...

It is given that $$f_n(x) = \sin x + \frac{1}{2}\sin 2x + \frac{1}{3}\sin 3x + \ldots + \left(\frac{...

Prove that, if $0 < x < 1$, \[\pi < \frac{\sin\pi x}{x(1-x)} \leq 4.\]...

Show that $e^{-t^2/2} \geq \cos t$ for $0 \leq t \leq \frac{1}{4}\pi$....

The function $\log^+ (x)$ is defined by \[\log^+ (x) = \begin{cases} \log_e (x) & (x \geq 1) \\ 0 &...

Prove that the positive number $a$ has the property that there exists at least one positive number t...

Let $$f_m(x) = \frac{x}{2} \left[ \sin x - \frac{\sin 2x}{2} + \frac{\sin 3x}{3} - \ldots + (-1)^{m+...

Find the ranges of values of $x$ for which the function $(\log x)/x$ (i) increases, (ii) decreases, ...

The polynomial $P(x)$ is defined, for a given positive integer $n$, by \[ P(x) = \frac{d^n y}{dx^n},...

$\alpha$ is a real number and \[ \frac{\alpha x - x^3}{1+x^2} \] is increasing for all real $x$. Sho...

Find for what ranges of $x$ the function $\dfrac{\log x}{x}$ increases as $x$ increases, and decreas...

Define $\log_e x$ for $x>0$. Prove that for $x>1$: \[ x^2-x > x\log_e x > x-1 \quad \text{an...

Prove that, if $a$ is real, the equation \[ e^x = x + a \] has two real roots if $a$ is greater than...

\begin{enumerate} \item[(i)] Prove that, for positive values of $x$, ...