The tangents at two points \(A\), \(A'\) of a circle \(S\) meet in \(T\). The mid-points of \(TA\), \(TA'\) are \(L\), \(L'\). \(P\) is any point of \(S\) and the lines \(PA\), \(PA'\) meet the line \(LL'\) in \(Q\), \(Q'\). Prove that \(QT^2 = QA \cdot QP\) and that the points \(T\), \(P\), \(Q\), \(Q'\) are concyclic.
\(OA\), \(OB\), \(OC\) are three lines through the point \(O\). The angles \(BOC\), \(COA\) and \(AOB\) are, respectively, \(\alpha\), \(\beta\) and \(\gamma\). Calculate \(\cos^2\theta\), where \(\theta\) is the angle between the line \(OA\) and the plane \(BOC\).
Points \(X\) and \(Y\) are chosen, on the perpendiculars (produced if necessary) from the vertices \(A\) and \(B\) of a triangle \(ABC\) to the opposite sides, so that \(AX = BC\) and \(BY = AC\). Prove that \(XOY\) is a right-angled isosceles triangle.
\(ABC\) is an isosceles triangle, with \(AB = AC\), \(I\) is the centre of the inscribed circle. \(S, I_1\) is the centre of the circle \(S_1\) touching \(BC\) internally and \(AB\), \(AC\) externally. Prove that the circle \(S'\) on \(I_1\) as diameter touches \(AB\), \(AC\) at \(B\) and \(C\), and that if \(S'\) meets \(S\) in \(P, Q\) and \(S_1\) in \(P_1, Q_1\), then \(I_1P\), \(I_1Q\) touch \(S\) and \(IP_1\), \(IQ_1\) touch \(S_1\).
A cube stands on a horizontal surface, and supports a second cube of equal size which is balanced on a vertex in such a way that its corresponding diagonal is vertical, and would if continued pass through the centre of the lower cube. The sun shines vertically overhead. Show that the upper cube can be rotated about its vertical diameter so that the lower cube will lie entirely in its shadow.
\(ABC\) is a non-isosceles triangle, with \(M\) the mid-point of \(BC\). A line passes through \(A\), \(B\) in \(P\), \(Q\) respectively, and \(AP = AQ\). Prove that \(BP = CQ\). If \(ABC\) is a non-isosceles triangle with points \(B\), \(C\), \(P\), \(Q\) cannot be concyclic, and that the triangles \(BMP\), \(CMQ\) cannot have equal areas.
A straight line meets the sides \(BC\), \(CA\), \(AB\) of a triangle \(ABC\) in \(L\), \(M\), \(N\) respectively. Prove that \(\frac{BL \cdot CM \cdot AN}{LG \cdot MA \cdot NB} = -1,\) due regard being paid to sign. The mid-points of the sides \(PQ\), \(RS\) of a parallelogram \(PQRS\) are \(X\), \(Y\) respectively. \(H\) is a point on the diagonal \(PR\) and \(HX\), \(HY\) meet \(QR\), \(PS\) respectively in \(U\), \(V\). Prove that \(UV\) is parallel to \(PQ\). If \(UV\) cuts \(PR\) in \(W\) prove that \(\frac{2}{HW} = \frac{1}{HP} + \frac{1}{HR},\) due regard being paid to sign.
A fixed point \(K\) lies inside a triangle \(ABC\) and a circle through \(A\) and \(K\) meets \(AB\), \(AC\) again in \(R\), \(Q\) respectively. Prove that the circles \(BRK\) and \(CQK\) meet again in a point \(P\) of \(BC\). Show further that all triangles \(PQR\) obtained in this way are similar, and that the one of smallest area has its vertices at the feet of the perpendiculars from \(K\) to the sides of \(ABC\).
A regular dodecahedron is bounded by twelve regular pentagons. Find to the nearest degree the obtuse angle between two adjacent faces.
\(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\) are consecutive vertices of a regular polygon of \(n\) sides (\(n \geq 7\)); \(BE\) meets \(DG\) in \(X\), \(CF\) in \(Y\), and \(AD\) in \(Z\). Prove that \(EX \cdot EZ = EY^2\).
A square \(ABCD\) is such that \(A\) lies on \(y = 0\), \(C\) on \(x = 0\), while \(B\) and \(D\) lie on the circle \(x^2 + y^2 + 2gx + 2fy = 0\) (\(f\), \(g\), \(f \neq g\) all non-zero). Show that three squares satisfy these conditions, and that any pair of these have a common vertex.
\(ABCDE\) is a regular pentagon of side 1. \(BD\) and \(CE\) meet in \(A'\), and \(DA\) and \(BC\) meet in \(C'\). Find the length of \(A'C'\). (Your answer should not contain trigonometric functions, but may contain square roots.)
Each of three circles \(C_1\), \(C_2\) and \(C_3\) meets the other two, but they do not have a common interior area. Show that there is a circle which meets each of them orthogonally. What happens if the three circles have a common area?
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 N. Prove that MN is perpendicular to AB, and that the circle on MN as diameter cuts the semi-circle orthogonally.
Find the locus of a point P which moves in a plane containing three distinct fixed points \(A_1\), \(A_2\), \(A_3\) subject to the restriction that \(PA_1^2 + PA_2^2 + PA_3^2\) is constant. Show that there is a value of this constant for which the locus reduces to a single point, and identify this point. If \(B_1\), \(B_2\), \(B_3\) are three other fixed points of the plane, prove that the locus of a point P which moves in the plane so that \(PA_1^2 + PA_2^2 + PA_3^2 = PB_1^2 + PB_2^2 + PB_3^2\) is in general a straight line. Discuss possible exceptional cases.
\(ABC\) is a triangle, and \(BCA', CAB', ABC'\) are equilateral triangles; \(A, A'\) being on opposite sides of \(BC\), \(B, B'\) on opposite sides of \(CA\) and \(C, C'\) on opposite sides of \(AB\). Prove that the lines \(AA', BB', CC'\) are of equal length and meet in a point.
The tangents at points \(A\) and \(B\) of a circle \(\Gamma\) meet at a point \(O\). A chord of \(\Gamma\) passes through \(O\) and intersects \(\Gamma\) at \(P\) and \(Q\). The lines \(AB\) and \(PQO\) meet at \(R\). Prove that \[\frac{1}{OP} + \frac{1}{OQ} = \frac{2}{OR}.\]
A solid is constructed by cutting the corners off a cube in such a way that its set of faces consists of six identical squares and eight identical equilateral triangles. At each vertex two triangular faces and two square faces meet. Find the cosine of the angle \(\theta\) between any two triangular faces which meet at a vertex.
Let \(C_1\), \(C_2\) and \(C_3\) be circles in the plane, each pair of which intersect in two points. The common tangents to \(C_2\) and \(C_3\) meet at \(P_1\), and points \(P_2\) and \(P_3\) are defined similarly. Prove that \(P_1\), \(P_2\) and \(P_3\) are collinear. What is the analogous result if the circles are mutually disjoint?
A triangle \(ABC\) has area \(\Delta\), and \(P\) is an interior point. The line through \(P\) parallel to \(BC\) cuts \(AB\) in \(W\) and \(AC\) in \(T\); the line through \(P\) parallel to \(CA\) cuts \(AB\) in \(V\) and \(BC\) in \(S\); and the line through \(P\) parallel to \(AB\) cuts \(AC\) in \(U\) and \(BC\) in \(R\). The triangles \(PBC\), \(PCA\), \(PAB\) have areas \(\alpha\), \(\beta\), \(\gamma\) respectively and the triangles \(AVU\), \(BWR\), \(CST\) have areas \(\alpha'\), \(\beta'\), \(\gamma'\) respectively. By first showing that $$\frac{\beta'}{\gamma} = \frac{\alpha}{\Delta},$$ prove that the area of the hexagon \(RSTUVW\) is at least \(\frac{9}{8}\Delta\). [You may assume that if positive numbers \(p\), \(q\), \(r\) add up to 1, then \(pq +qr+rp\) cannot exceed \(\frac{1}{3}\).]
Show that it is not always possible to inscribe a circle within a convex quadrilateral with sides (taken consecutively) of given lengths, but that if it is possible then a circle may be inscribed in any convex quadrilateral with sides (taken consecutively) of these lengths.
Given a triangle \(ABC\) show that it is possible to construct three mutually touching circles with centres \(A\), \(B\), \(C\), respectively. What values are possible for the radius of the circle centre \(A\) in terms of the lengths \(a\), \(b\), \(c\) of the sides of the triangle?
The great grey green greasy Limpopo river is 1 kilometre wide and flows with negligible speed between parallel banks. A young elephant wishes to reach a fever tree \(h > 0\) kilometres upstream on the other side as quickly as possible. He gallops a distance \(x \geq 0\) kilometres (upstream) and then plunges in and swims directly to the tree. If he gallops at a speed of \(v\) kilometres an hour and swims at a speed of \(u\) kilometres an hour \((v \geq u > 0)\) what value of \(x\) should he choose? Explain briefly why the character of the solution is different for large and small \(h\).
Show that angles subtended by a chord of a circle at the circumference and in the same segment are equal. A rod is bent so as to form an acute angle at \(X\). Another rod \(PQ\) slides with its ends \(P\) and \(Q\) on the two straight arms of the bent rod. At each position of \(P\) and \(Q\) lines \(PR\), \(QR\) are drawn perpendicular to the arms on which respectively \(P\) and \(Q\) move. Show that, when the bent rod is fixed and \(PQ\) moves, \(R\) moves on a circle. Show further that, when \(PQ\) is fixed and the bent rod is moved, \(R\) again moves on a circle, of radius half that of the former circle.
Let \(ABCDE\) be a regular pentagon and let \(AC\) and \(BE\) intersect at \(H\). Prove that \(AB = CH = EH\) and that \(AB\) is tangent to the circle \(BHC\).
The surfaces of two spheres have more than one real common point. Prove that they intersect in a circle. A triangle \(BCD\) is given in a plane \(\alpha\). Prove that there are just two possible positions, reflections of each other in \(\alpha\), for a point \(A\), which is such that the angles \(BAC\), \(CAD\), \(DAB\) are right angles, if and only if the triangle \(BCD\) is acute-angled. Find the distance of \(A\) from \(\alpha\) in the case where \(BCD\) is an equilateral triangle with sides of unit length.
Two regular tetrahedra are formed from among the vertices of a cube of edge length \(a\). Find the volume of the portion of the cube external to both tetrahedra.
The centres of two large solid hemispherical radar domes of radii \(a\) and \(b\) are at a distance \(c\) apart. An aesthete wishes to stand at the point, on the line of centres between the two hemispheres, at which the least amount of hemispherical surface area is visible. Where should he stand?
State Pythagoras's Theorem. Two circles \(\alpha\), \(\beta\) with centres \(A\) and \(B\) and radii \(a\) and \(b\), lie in different planes \(\pi\) and \(\varpi\) respectively which meet in a line \(l\). Show that the two circles will lie on the same sphere if and only if \(AB\) is perpendicular to \(l\) and \[AP^2-BP^2 = a^2-b^2\] for every point \(P\) on \(l\).
A solid fills the region common to two equal circular cylinders whose axes meet at right angles. Prove that its volume is \(4/\pi\) times the volume of a sphere with radius equal to that of the cylinders.
A hill \(\frac{1}{2}\) mile high is in the shape of a spherical cap, with a horizontal circular rim, the radius of the sphere being 1 mile. A man walks up from a point of the rim to the peak at a steady speed of 3 miles per hour but never ascending at a gradient of more than \(\sin^{-1}(\frac{1}{3})\). Find the minimum time the walk can take him, and sketch roughly a possible minimum path (as seen from above); does it necessarily have no sharp corners?
Let \(A'\) be a point in the plane of a triangle \(BCD\). Let \(BC\) and \(A'D\) meet at \(X\), and \(A'B\) meet at \(Y\), and let \(l\) denote the line \(XY\). Let \(l\) meet \(BD\) and \(A'C\) at \(P\) and \(Q\) respectively. Prove that \(P\) and \(Q\) divide \(XY\) internally and externally in the same ratio (i.e. that \(\frac{XP}{PY} = -\frac{XQ}{QY}\)). Suppose that \(ABCD\) is a tetrahedron and that \(A'\) is a general point in the plane of the face \(BCD\). Show that it is possible to find points \(B'\), \(C'\) and \(D'\) so that the two tetrahedra \(ABCD\) and \(A'B'C'D'\) have the property that the plane of any face of either tetrahedron contains precisely one vertex of the other. Are \(B'\), \(C'\) and \(D'\) uniquely determined?
Let \(\cal S\) be an infinite set of pairs of points in the plane such that the points in question do not all lie on a circle or on a straight line. Show that the following two conditions on \(\cal S\) are equivalent: (i) there is a fixed point \(P\) and a constant \(k\) such that for all pairs \(\{X, X'\}\) in \(\cal S\), \(P\) lies on the line segment \(XX'\) and \(XP.PX' = k^2\); (ii) the four points of any two pairs in \(\cal S\) lie on a circle or line, in an ordering in which the pairs are interleaved. \(T\) is a transformation of the plane \(\Pi\), with an inverse \(T^{-1}\), such that both \(T\) and \(T^{-1}\) send all circles to circles. Let \(\mathcal{C}(P, k)\) be the set of all circles in \(\Pi\) containing a chord \(XX'\) such that \(P\) lies on the segment \(XX'\) and \(XP.PX' = k^2\). Show that \(T\) maps \(\mathcal{C}(P, k)\) to a set of circles of the form \(\mathcal{C}(\tilde{P}, \tilde{k})\). Show that \(T\) can be extended to a transformation of three-dimensional space containing \(\Pi\), that maps all spheres with centres in \(\Pi\) to spheres with centres in \(\Pi\).
A regular tetrahedron, with edges of length \(a\), is inscribed in a sphere of radius \(R\). Find the value of the ratio \(a/R\).
\(ABCD\) is a tetrahedron. \(O\) is a point not lying on any of its faces. The line through \(O\) and \(A\) cuts \(BCD\) in \(P\) and \(P'\), respectively. Similarly the line through \(O\) meeting \(BC\) and \(AD\) cuts them in \(Q\) and \(Q'\) respectively, and that meeting \(AB\) and \(CD\) cuts them in \(R\) and \(R'\), respectively. Prove that \(AP\), \(BQ\), \(CR\) and \(DO\) are concurrent.
Four points \(P_1\), \(P_2\), \(P_3\), \(P_4\) are not coplanar. The line through \(P_1\) and \(P_3\) is denoted by \(l_{13}\). A plane \(\pi\) cuts \(l_{13}\) in \(Q_{13}\). Prove that \(Q_{13}\), \(Q_{24}\), \(Q_{14}\) are collinear. Draw the figure in the plane \(\pi\) that is the intersection of \(\pi\) with the edges and faces of the tetrahedron \(P_1 P_2 P_3 P_4\). Describe the effect on the figure if \(\pi\), \(P_1\), \(P_3\) are kept fixed, and \(P_2\), \(P_4\) are moved to different positions on \(l_{24}\), \(l_{24}\), respectively.
Take any two of the standard concurrence theorems for the triangle (medians, altitudes, bisectors of angles, perpendicular bisectors of sides), formulate analogous theorems in three dimensions, with `triangle' replaced by `tetrahedron', and discuss whether or not the three-dimensional theorems are true.
Find the \emph{width} of a regular tetrahedron of side \(a\), where \emph{width} is defined as the least distance between a pair of distinct parallel planes, each of which 'touches' the tetrahedron, either at a vertex, along an edge, or on a face. Hence or otherwise show that it is impossible to put a regular tetrahedron of side greater than \(\sqrt{2}\) in a cube of side 1. Show, with a sketch, how a regular tetrahedron of side exactly \(\sqrt{2}\) may be put in such a cube (the boundary of the cube may be used).
\(O\) is a point inside a convex polygon \(ABC\ldots N\), of \(n\) sides; \(A_1, B_1, C_1, \ldots, N_1\) are the feet of the perpendiculars from \(O\) on to the sides \(AB, BC, CD, \ldots, NA\) respectively. The process is repeated for the same point \(O\) and the polygon \(A_1B_1C_1\ldots N_1\), to get a polygon \(A_2B_2C_2\ldots N_2\), and is continued, to get in succession polygons \(A_iB_iC_i\ldots N_i\), \(i = 3, 4, \ldots\). Prove that the polygon \(A_nB_nC_n\ldots N_n\) is similar to the original polygon \(ABC\ldots N\). Examine whether, in the case \(n = 3\), the result remains true if \(O\) is exterior to the triangle, mentioning any special cases that may arise.
A tetrahedron \(ABCD\) is such that there is a sphere which touches its six edges. Prove also that the three sums of pairs of opposite edges are the same. Prove also that the three loci of points of contact of opposite edges are concurrent.
A cube of side \(2a\) has horizontal faces \(ABCD\), \(A'B'C'D'\) and vertical edges \(AA'\), \(BB'\), \(CC'\), \(DD'\). Find the radius of the sphere inscribed in the tetrahedron \(AB'CD'\).
Prove that there exists a sphere touching the six edges of the tetrahedron \(ABCD\) internally if, and only if, \(AB + CD = AC + BD = AD + BC\). If, in addition, there exists a sphere touching \(CD\), \(DB\), \(BC\) internally and also touching \(AB\), \(AC\), \(AD\) produced, prove that \(AB = AC = AD\), and that the triangle \(BCD\) is equilateral.
From a point \(O\) perpendiculars \(OA'\), \(OB'\), \(OC'\), \(OD'\) are drawn to the faces of a tetrahedron \(ABCD\). Prove that pairs of lines such as \((AB, C'D')\), \((BC, A'D')\) are mutually perpendicular. Hence prove that any pair of perpendiculars from \(A\), \(B\), \(C\), \(D\) to the corresponding faces of the tetrahedron \(A'B'C'D'\) are coplanar, and deduce that all these perpendiculars are concurrent.
A convex polyhedron \(P\) has, for its faces, \(x\) triangles and \(y\) (convex) quadrilaterals, where \(x\) and \(y\) are both positive. The faces meeting in a vertex of \(P\) consist of \(a\) triangles and \(b\) quadrilaterals, where \(a\) and \(b\) are the same for all vertices. Prove that $$\frac{3x}{a} = \frac{4y}{b} = \frac{1}{2}(x + 2y + 4),$$ and find all the positive integral values of \(a\), \(b\), \(x\), \(y\) that satisfy these equations. Give examples of polyhedra satisfying these conditions when \(b = 2\).
Prove that the area of a sphere \(S\) between two parallel planes \(\pi\), \(\pi'\) both of which meet \(S\) depends only on the radius of \(S\) and the distance between \(\pi\) and \(\pi'\). A circular disc \(D\) of radius 1 may be covered by \(r\) planks each of width \(w\). By considering the areas above the planks on the sphere of which \(D\) is the equator, show that \(rw \geq 2\), and that if \(rw = 2\) then the planks must be placed parallel to one another.
A tetrahedron \(ABCD\) is given; \(L\), \(M\), \(N\) are the middle points of \(BC\), \(CA\), \(AB\) respectively and \(U\), \(V\), \(W\) are the middle points of \(AD\), \(BD\), \(CD\). Prove that the lines \(LU\), \(MV\), \(NW\) have a common middle point \(O\). Prove that, if \(LU\), \(MV\), \(NW\) are mutually perpendicular, then the four faces of the tetrahedron are congruent acute-angled triangles.
In a tetrahedron \(ABCD\) the edges \(AD\) and \(BC\) are perpendicular, \(AB = CD\), and \(AC = BD\). Prove that \(AB = AC\).
Prove that in general the perpendicular from a vertex on to the opposite face of a tetrahedron is intersected by the straight lines drawn through the orthocentre of, and perpendicular to, the other three triangular faces. Explain what happens if (i) one pair of opposite edges are perpendicular, and (ii) two pairs of opposite edges are perpendicular. In the latter case prove that the centroid of the tetrahedron is the midpoint of the join of the centre of the circumscribing sphere and the meet of the perpendiculars from the vertices on to the opposite faces.
Show that if the distance between the points \(A\) and \(B\) is greater than \(d\), then the two spheres of radius \(\frac{1}{2}d\) with centres at \(A\) and \(B\) have no point in common. Hence show that if \(P_1, \ldots, P_n\) are \(n\) points contained in a cube of side \(X\), such that the distance between each pair of these points is at least \(1\), then $$n < \frac{6(X+1)^3}{\pi}.$$
Given any four points on the surface of a sphere of unit radius, prove that it is possible to find two of them whose distance apart is at most \(\sqrt{3}\).
\(ABCD\) is a square, whose opposite vertices \(A,C\) lie, respectively, on the lines \(y = mx, y = -mx\). If the equation of \(AC\) is \(\frac{x}{a} + \frac{y}{b} = 1\), find the coordinates of \(B\) and \(D\). Hence, or otherwise, show that if \(AC\) varies in such a way that \(B\) lies on the line \(px + qy + r = 0\), then the locus of \(D\) is a straight line, and find its equation.
Let \(n\) be a positive integer. What is the largest number \(M\) of maxima that the polynomial \[f(x) = x^n+a_1x^{n-1}+ \ldots +a_{n-1}x+a_n\] can have? For each \(n\), give an example of a polynomial which has \(M\) maxima, and justify your answer.
A farmer wishes to provide his cattle with three nutrients \(A, B\) and \(C\), for which the minimum requirements of 21, 9 and 12 units respectively. Two animal foods \(F_1\) and \(F_2\) are available; their content for unit cost are given in the following table.
Discover all the real roots of each of the equations
Let \(J_1\) be the operation of taking the inverse (reciprocal) of a number, and \(J_2\) the operation of subtracting a number from 1. Prove that the operations \(J_1\) and \(J_2\), applied repeatedly in any order to a number \(\lambda\) (\(\lambda \neq 0, \lambda \neq 1\)) can only lead to one of a finite set of numbers. Express each of these numbers in terms of \(\lambda\). If \(J_r\) is the operation by which the \(r\)th member of the set (excluding \(\lambda\) itself) is obtained from \(\lambda\), show that either \(J_r\) applied twice reproduces the original number, or \(J_r\) does so when applied three times.
Determine \(\theta\) so that the line \[lx + my + n = \theta(l'x + m'y + n')\] is perpendicular to the line \[\lambda x + \mu y + \nu = 0.\] Prove that the perpendiculars from the vertices \(A_1, A_2, A_3\) of the triangle formed by the lines \[l_ix + m_iy + n_i = 0 \quad (i = 1, 2, 3)\] on to the sides \(B_2B_3, B_3B_1, B_1B_2\), respectively, of the triangle formed by the lines \[\lambda_ix + \mu_iy + \nu_i = 0 \quad (i = 1, 2, 3)\] are concurrent if \[(l_1\lambda_1 + m_1\mu_1)(l_2\lambda_2 + m_2\mu_2)(l_3\lambda_3 + m_3\mu_3) = (l_1\lambda_2 + m_1\mu_2)(l_2\lambda_3 + m_2\mu_3)(l_3\lambda_1 + m_3\mu_1).\] Deduce that in this case the perpendiculars from the vertices of \(B_1B_2B_3\) on to the respective sides of \(A_1A_2A_3\) are also concurrent.
(i) Show that in rectangular cartesian coordinates the equation $$p(x^4 + y^4) + qxy(x^2 - y^2) + rx^2y^2 = 0$$ represents always two pairs of straight lines at right angles. Find the condition that the two pairs will coincide. (ii) Find the area of the triangle formed by the lines whose equations in rectangular cartesian coordinates are $$ax^2 + 2hxy + by^2 = 0$$ and $$lx + my + 1 = 0.$$
A right-angled triangle has integral sides and the lengths of the two shorter sides differ by 1. If the sides are \(m\), \(m+1\), \(n\), show that there exist right-angled triangles having the same property with hypotenuse \(3n \pm (4m + 2)\). Hence or otherwise, show that there are just four such triangles with hypotenuse less than 1000.
If \(x+y+z+t=0\), prove that
If \(a, b, c\) are unequal non-zero numbers, solve the simultaneous equations \begin{align*} x+y+z &= a+b+c, \\ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} &= 1, \\ \frac{x}{a^3} + \frac{y}{b^3} + \frac{z}{c^3} &= 0, \end{align*} distinguishing the various cases that may arise.
Solve the equations \begin{align*} x+y+z &= 3, \\ x^2+y^2+z^2+2z &= 9, \\ xyz+xy &= -2. \end{align*}
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 otherwise, establish the identity \[ (a^3+b^3+c^3-3abc)^2 = (a^2-bc)^3 + (b^2-ca)^3 + (c^2-ab)^3 - 3(a^2-bc)(b^2-ca)(c^2-ab). \]
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 pressure, specific volume and temperature satisfy the equation \(pv = Rt\), where \(R\) is a constant, then the difference between the specific heat at constant pressure and the specific heat at constant volume, measured in work units, is equal to \(R\).
Prove that if \[ a\frac{y+z}{y-z} = b\frac{z+x}{z-x} = c\frac{x+y}{x-y}, \] each of these expressions \(= \pm \left\{-\frac{abc}{a+b+c}\right\}^{\frac{1}{2}}\).
Solve the equations \[ \frac{1}{y} - \frac{1}{z} = a - \frac{1}{a}, \quad y - \frac{1}{z} = b - \frac{1}{c}, \quad z - \frac{1}{x} = c - \frac{1}{a}. \]
Show that if \(ax+by+cz=0\) for all values of \(x, y,\) and \(z\) such that \(\alpha x + \beta y + \gamma z = 0\), then \[ \frac{a}{\alpha} = \frac{b}{\beta} = \frac{c}{\gamma}. \]
Having given that \[ \frac{x^2-yz}{a} = \frac{y^2-zx}{b} = \frac{z^2-xy}{c}, \] prove that \[ \frac{a^2-bc}{x} = \frac{b^2-ca}{y} = \frac{c^2-ab}{z}; \] and shew that the product of one of the first three expressions and one of the last three is \[ (a+b+c)(x+y+z). \]
If \(l, m, l', m', l''\) and \(m''\) are integers, and if \(\alpha/\beta\) is not rational, and if \[ l\alpha + m\beta = l'\alpha + m'\beta, \] shew that \(l=l'\), and \(m=m'\). Also shew that no two of the numbers \[ (2l+5m)\alpha+l\beta, \quad (2l'+5m'+1)\alpha+l'\beta, \quad (2l''+5m'')\alpha+(l''+1)\beta \] are equal.
Prove that, if \(p/q\) is a fraction in its lowest terms, then integers \(r\) and \(s\) can be found such that \(qr-ps=1\). Prove that, if \(p/q\) and \(r/s\) are fractions such that \(qr-ps=1\), then the denominator of any fraction whose value lies between \(p/q\) and \(r/s\) is at least \(q+s\).
At an election the majority was 1184, which was one-fifth of the total number of votes; how many votes did each side poll?
An article when sold at a profit of 13 per cent. yields 1s. 5d. more profit than when sold at a profit of \(4\frac{1}{2}\) per cent.; find the prime cost of the article.
A figure of four triangles and three squares is constructed by describing squares P, Q, R externally on the three sides of a triangle and joining their adjacent corners. P contains 5 sq. cm., Q 10 sq. cm. and the areas of the triangles are together half of that of the squares. Prove that the square R contains either 5 or 13 sq. cm.
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^2+c^2-a^2) &= y(c^2+a^2-b^2), \end{align*} prove that \[ a^3+b^3+c^3 = (b+c)(c+a)(a+b). \]
Find all the real solutions of the equations:
Prove that the geometric mean of \(n\) positive numbers cannot exceed their arithmetic mean. Deduce that if \(x\), \(y\) and \(z\) are positive numbers such that $$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1,$$ then $$(x-1)(y-1)(z-1) \geq 8.$$
Positive rational 'weights' \(m_1, \ldots, m_n\) are attached to positive numbers \(a_1, \ldots, a_n\). Use the inequality connecting the arithmetic and geometric means to prove that \begin{align*} \frac{m_1a_1 + \ldots + m_na_n}{m_1 + \ldots + m_n} \geq (a_1^{m_1} a_2^{m_2} \ldots a_n^{m_n})^{1/(m_1 + \ldots + m_n)}. \end{align*} By attaching suitable weights to 1 and \(1 + x/n\), prove that, if \(x\) is positive, \begin{align*} \left(1 + \frac{x}{n+1}\right)^{n+1} \geq \left(1 + \frac{x}{n}\right)^n. \end{align*}
Prove that, if \(n > 1\), $$\sum_{r=1}^n \left(1 + \frac{1}{r}\right) > n(n+1)^{1/n}.$$ If you appeal to any general inequality, prove it. Prove that $$n\{(n+1)^{1/n} - 1\} < \sum_{r=1}^n \frac{1}{r} < n\left\{1 + \frac{1}{n+1} - \frac{1}{(n+1)^{1/n}}\right\}.$$
Prove that (if all the numbers involved are positive) $$(ab)^{\frac{1}{2}} \leq \frac{1}{2}(a+b) \quad \text{and} \quad (abcd)^{\frac{1}{4}} \leq \frac{1}{4}(a+b+c+d).$$ By taking a special value for \(d\), or otherwise, prove that $$(abc)^{\frac{1}{3}} \leq \frac{1}{3}(a+b+c).$$ Prove also that $$8abc \leq (b+c)(c+a)(a+b).$$
\(a\), \(b\), \(c\) are three positive numbers. Prove the inequality $$abc \geq (b + c - a)(c + a - b)(a + b - c).$$ Is the inequality $$abc \geq \kappa(b + c - a)(c + a - b)(a + b - c)$$ true for any constant \(\kappa\) greater than 1 (and all \(a\), \(b\), \(c\))?
Show SolutionProve that the geometric mean of a finite set of positive real numbers does not exceed their arithmetic mean. Prove that \(k = \frac{2\sqrt{2}}{3}\) is the smallest constant which has the following property: if \(a\), \(b\) are real numbers such that \(a \geq 2b > 0\), then \[ \sqrt{(ab)} \leq k\left(\frac{a+b}{2}\right). \] Show that, if \(a_1\), \(\ldots\), \(a_n\), \(b_1\), \(\ldots\), \(b_n\) are real numbers such that \[ a_i \geq 2b_i > 0 \quad (i = 1, \ldots, n), \] then \[ (a_1a_2\ldots a_nb_1b_2\ldots b_n)^{\frac{1}{2n}} \leq \frac{2\sqrt{2}}{3}\left(\frac{a_1 + a_2 + \ldots + a_n + b_1 + b_2 + \ldots + b_n}{2n}\right). \]
Prove that if \(x_1, x_2, \ldots, x_n\) and \(y_1, y_2, \ldots, y_n\) are two sets of positive quantities, both in increasing order of magnitude, then \[\frac{1}{n} \sum_{r=1}^{n} x_r y_r > \left(\frac{1}{n} \sum_{r=1}^{n} x_r\right) \left(\frac{1}{n} \sum_{r=1}^{n} y_r\right).\] Prove that if \(a\), \(b\), \(c\) are three unequal positive quantities, then \[a^3 + b^3 + c^3 > abc(a^2 + b^2 + c^2).\]
Let \(p_i\) (\(1 \leq i \leq n\)) and \(q_i\) (\(1 \leq i \leq n\)) be real numbers such that $$p_1 \geq p_2 \geq \ldots \geq p_n \geq 0$$ and $$q_1 + \ldots + q_i \geq i \quad (1 \leq i \leq n).$$ Show that $$p_1 q_1 + \ldots + p_n q_n \geq p_1 + \ldots + p_n.$$ Hence, or otherwise, show that if \(a_i\) (\(1 \leq i \leq n\)) and \(b_i\) (\(1 \leq i \leq n\)) are real numbers such that $$a_1 \geq a_2 \geq \ldots \geq a_n > 0$$ and $$b_1 b_2 \ldots b_i \geq a_1 a_2 \ldots a_i \quad (1 \leq i \leq n),$$ then $$b_1 + \ldots + b_n \geq a_1 + \ldots + a_n.$$
Prove that, if \(ax^2+2bx+c\) is to be positive for all real values of \(x\), it is both necessary and sufficient that
Two polynomials, \(P\) and \(Q\), have no factor in common. Shew that the maximum and minimum values of \(P/Q\) are the values of \(\lambda\) for which \(P-\lambda Q=0\) has a real root of even multiplicity. \par Shew further that the value \(\lambda\) is a maximum or a minimum according as \[ \{P^{(2v)}(\alpha)Q(\alpha) - P(\alpha)Q^{(2v)}(\alpha)\} \le 0, \] \(\alpha\) being the root of \(P-\lambda Q=0\) of multiplicity \(2v\). \par Find the turning values of \[ \frac{x+2}{x^2+x+2}, \] distinguishing between maxima and minima. \par [The condition that the cubic \(y^3+gy+h=0\) should have a pair of equal roots is that \(4g^3+27h^2=0\).]
Find the condition that \(ax+b/x\) can take any real value for real values of \(x\). Express \(\xi = (x-a)(x-b)/(x-c)(x-d)\) in terms of \(y\) where \(y=(x-d)/(x-c)\), and hence or otherwise shew that \(\xi\) can take all real values if \((c-a)(c-b)(d-a)(d-b)\) is negative.
Shew that two quadratic expressions \(ax^2+2bx+c\) and \(a'x^2+2b'x+c'\) can generally be expressed in the forms \(p(x-\alpha)^2+q(x-\beta)^2\) and \(r(x-\alpha)^2+s(x-\beta)^2\) respectively. Find a function \(\frac{ax^2+2bx+c}{a'x^2+2b'x+c'}\) which has turning values 3 and 4 when \(x=2\) and \(-2\) respectively, and has the value 6 when \(x=0\).
Shew how to express \(ax^2+2bx+c\) and \(a'x^2+2b'x+c'\) simultaneously in the forms \(p(x-\alpha)^2+q(x-\beta)^2\) and \(p'(x-\alpha)^2+q'(x-\beta)^2\). Apply your method to \[ 7x^2-22x+28 \quad \text{and} \quad 27x^2-62x+68. \]
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 positive, provided that any two of the numbers \(a, b, c\) are together greater than the third.
Simplify \(\dfrac{2x+5}{6x+7} - \dfrac{2x-1}{6x+5} - \dfrac{32x+33}{36(x+1)^2-1}\). and prove that, if \(a+b+c=0\), then \(a^3+b^3+c^3=3abc\).
Obtain the condition for the equation \(ax^2 + 2bx + c = 0\) to have real roots, where \(a\), \(b\) and \(c\) are real numbers. The real numbers \(p\), \(q\) and \(r\) are such that none has unit modulus, and \(p^2 + q^2 + r^2 + 2pqr = 1.\) Prove that \(p\), \(q\), and \(r\) either all lie between \(+1\) and \(-1\), or all lie outside this range.
Let \(f(x) = ax^2 + bx + c\) (\(a\), \(b\), \(c\) real, \(a > 0\)). Explain why the following statements are equivalent. (i) \(f(x) \leq 0\) for some real number \(x\). (ii) \(b^2 - 4ac \geq 0\). The real numbers \(a_1\), \(a_2\), ..., \(a_n\), \(b_1\), \(b_2\), ..., \(b_n\) are such that \(b_1^2 - b_2^2 - ... - b_n^2 > 0\). By considering the expression \((b_1 x - a_1)^2 - (b_2 x - a_2)^2 - ... - (b_n x - a_n)^2\), or otherwise, prove that \((a_1^2 - a_2^2 - ... - a_n^2)(b_1^2 - b_2^2 - ... - b_n^2) \leq (a_1 b_1 - a_2 b_2 - ... - a_n b_n)^2\).
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. Deduce that if \(\alpha\), \(\beta\), \(\gamma\) are real numbers then \[(a^4+\beta^4+\gamma^4)(a^2+\beta^2+\gamma^2) \geq (a^2+\beta^2+\gamma^2)^2.\] Give necessary and sufficient conditions on \(\alpha\), \(\beta\), \(\gamma\) for equality to hold.
Let \(a, b, c\) be integers and let \(f(x, y) = ax^2 + 2bxy + cy^2\). Show that there are integers \(p, q, r, s\) such that \(ps - qr = 1\) and \(f(x, y) = 2(px + qy)(rx + sy)\) if and only if \(a\) and \(c\) are even and \(b^2 - ac = 1\).
Prove that, if \(a\), \(b\), \(h\) are real numbers such that \(a > 0\), \(ab - h^2 > 0\), then \[ax^2 + 2hx + b > 0\] for all real values of \(x\). If \(p\), \(q\), \(r\) are real, investigate the conditions under which \[px^2 + 2qx + r > \rho\] for all real values of \(x\). Show that these conditions imply that \(r > 2|q|\).
If \[x_1x_2 + y_1y_2 = a_1, \quad x_2x_3 + y_2y_3 = a_2, \quad x_3x_1 + y_3y_1 = a_3, \quad x_1^2 + y_1^2 = x_2^2 + y_2^2 = x_3^2 + y_3^2 = b,\] prove that \[b^3 - (a_1^2 + a_2^2 + a_3^2)b + 2a_1a_2a_3 = 0.\] Deduce that, if the \(2n\) equations \[x_1x_2 + y_1y_2 = a_1, \quad x_2x_3 + y_2y_3 = a_2, \ldots, x_{n-1}x_n + y_{n-1}y_n = a_{n-1}, \quad x_nx_1 + y_ny_1 = a_n,\] \[x_1^2 + y_1^2 = x_2^2 + y_2^2 = \ldots = x_n^2 + y_n^2 = b,\] for the \(2n\) unknowns \(x_1, \ldots, x_n, y_1, \ldots, y_n\), are consistent, there must be an algebraic relation connecting \(b\) and \(a_1, a_2, \ldots, a_n\).
Determine the limitations, if any, on the value of \(p\) if the expression $$x^2(y^2 + 2y + 2) + 2x(y^2 + 2py + 2) + (y^2 + 2y + 2)$$ is greater than or equal to zero for all pairs of real values of \(x\) and \(y\).
Prove that the expression \[5x^2 + 6y^2 + 7z^2 + 2yz + 4zx + 10xy\] is positive for all real values of \(x\), \(y\), \(z\), other than \(x = 0\), \(y = 0\), \(z = 0\). Find a set of real values of \(x\), \(y\), \(z\) for which the expression \[5x^2 + 6y^2 + 7z^2 - 2yz - 4zx - 10xy\] is negative. (If you quote a general test in support of your arguments, you must prove it.)
What conditions on the real numbers \(a\), \(b\), \(c\) are needed to ensure that \begin{align} \frac{ax^2 + bx + c}{cx^2 + bx + a} = \lambda \end{align} has a real root \(x\) for every real \(\lambda\)?
Show SolutionProve that, if \(a>0\) and \(ac-b^2>0\), then \(ax^2+2bx+c > 0\) for all real values of \(x\). Examine whether it is possible to find real values of \(x, y\) and \(z\) which give a negative value to the expression \[ 7x^2+10y^2+z^2-6yz+zx-8xy. \]
If \(\alpha, \beta\) denote the roots of a given quadratic equation \(Ax^2+Bx+C=0\), find the quadratic of which the roots are \(\frac{a\alpha^2+b\alpha+c}{a'\alpha^2+b'\alpha+c'}\) and \(\frac{a\beta^2+b\beta+c}{a'\beta^2+b'\beta+c'}\). Prove that, if \(x\) be restricted to be real, \(\frac{kx^2+kx+1}{x^2+kx+k}\) can have all values in case \(k\) is negative and not numerically less than \(\frac{1}{4}\); that there are two values between which it cannot lie when \(k\) is negative and numerically less than \(\frac{1}{4}\), or also when \(k>4\); and that there are two values between which it must lie in case \(k\) is positive and less than 4, these two values being coincident when \(k=1\).
A train travels 525 miles; if its average rate had been \(2\frac{1}{2}\) miles per hour faster, it would have taken 1 hour less; find its average rate.
Find the conditions that the equation \(ax^2+2bx+c=0\) should have (i) both its roots positive and (ii) two equal roots.
Given that \[xy - 3x - 2y + 4 = 0,\] evaluate \[\frac{(x-1)(y-4)}{(x-4)(y-1)}.\] If also \[xz - 6x - z + 8 = 0,\] find numbers \(p\), \(q\) such that \[\frac{(y-p)(z-q)}{(y-q)(z-p)}\] is a numerical constant, to be evaluated.
Solve the equations \begin{align} x + y^3 + z^3 &= 0,\\ x^3 + y + z^3 &= 0,\\ x^3 + y^3 + z &= 0, \end{align} given that no two of \(x, y, z\) are equal.
Prove that, if \(x_1\) and \(x_2\) are connected by the relation $$ax_1x_2 + bx_1 + cx_2 + d = 0,$$ there are, in general, two unequal values \(m\), \(n\) of \(x_1\) for which \(x_2 = x_1\); and that the relation is equivalent to $$\frac{x_2 - m}{x_2 - n} = k \frac{x_1 - m}{x_1 - n},$$ where \(k\) is a root of the equation $$(bc - ad)(k^2 + 1) + (b^2 + c^2 - 2ad)k = 0.$$ Find the relation between \(a\), \(b\), \(c\), \(d\) in order that the equations \begin{align} ax_1x_2 + bx_1 + cx_2 + d &= 0,\\ ax_1x_3 + bx_2 + cx_3 + d &= 0,\\ ax_2x_4 + bx_3 + cx_4 + d &= 0,\\ ax_3x_4 + bx_4 + cx_1 + d &= 0 \end{align} may be satisfied by values of \(x_1\), \(x_2\), \(x_3\), \(x_4\) which are all different.
If \(x\), \(y\), \(z\) are all different and \(x + \frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x},\) prove that the common value of these three expressions is \(\pm 1\).
Given that \(s^2 + c^2 = 1\), prove that \[(4c^3 - 3c)^2 + (4s^3 - 3s)^2 = 1.\] Given, conversely, that \[(4c^3 - 3c)^2 + (4s^3 - 3s)^2 = 1,\] prove that \(s\), \(c\) satisfy one or other of three relations of the form \[ps^2 + qsc + rc^2 = 1,\] where \(p\), \(q\), \(r\) are numbers (not necessarily rational), to be determined.
Show that, when \(a,b\) and \(c\) are real and positive, the system of equations \begin{equation} \tag{1} xyz = a(y+z) = b(z+x) = c(x+y) \end{equation} has real non-zero solutions \(x,y,z\) if and only if there is a proper triangle whose sides are proportional to \((a^{-1}, b^{-1}, c^{-1})\), and find the complete solution of the system in this case. Show also that, when \(b=1\) and \(c=-1\), the system \((1)\) has real non-zero solutions if and only if either \(a > \frac{1}{2}\) or \(-\frac{1}{2} < a < 0\).
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 putting \(y=tx\); and show that the parameters \(t_1, t_2, t_3\), of three collinear points satisfy the relation \[ t_2 t_3 + t_3 t_1 + t_1 t_2 + 1=0. \] The tangent at a point \(P\) meets the curve again in a point \(Q\): express the parameter of the point \(Q\) in terms of that of \(P\) and hence show that two straight lines \(QP_1, QP_2\) can be drawn from any point \(Q\) of the curve to touch it other than the tangent at \(Q\). Show also that as \(Q\) varies, the chord \(P_1P_2\) touches the conic \(y^2+8(x+1)(x+2)=0\).
Solve the equations
Solve the equation
If \[ x(1+\sin^2\phi-\cos\phi) = (y\sin\phi+a)(1+\cos\phi) \] and \[ y(1+\cos^2\phi) = (x\cos\phi+a)\sin\phi, \] prove that \[ y^2=ax. \]
Solve the equations:
Solve the equations:
Solve the equations:
Solve for \(x, y, z\) in terms of \(p, q, r\) the simultaneous equations \begin{align*} x+y+z &= 1, \\ x+py+3z &= 2, \\ x+y+qz &= r, \end{align*} and obtain the solutions in the special cases: (i) \(p=1, q \ne 1\), (ii) \(p \ne 1, q=1\), (iii) \(p=q=1\), giving any additional condition required for the existence of the solutions.
Solve the equations:
Find the equation of the tangent at a point on the curve given by \[ x/t = y/t^2 = 3a/(1+t^3). \] If the tangents at the points whose parameters are \(t_1, t_2, t_3, t_4\) are concurrent, prove that \[ \sum_{r=1}^4 t_r = 2 \prod_{r=1}^4 t_r. \]
Show that, for every positive integer \(n\), the number \(n^9 - n\) is divisible by 30 and that, for every odd positive integer \(n\), \(n^9 - n\) is divisible by 480.
The letters \(n\) and \(k\) denote positive integers.
Suppose that \(n\), \(x\) and \(y\) are positive integers such that \(n+x\) is a square and \(n+y\) is the next larger square. Show that \(n+xy\) and \(n+xy+x+y\) are adjacent squares. Hence show that \(n+x+y+xy(2+x+y+xy)\) is a square.
Show SolutionShow that if \(a, b, c, d \in \mathbb{Q}\), the rational numbers, and \(a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} = 0\), then \(a = b = c = d = 0\). Let \(V\) be the set of all real numbers of the form \(a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}\), where \begin{equation*} a, b, c, d \in \mathbb{Q}. \end{equation*} Show that if \(\alpha, \beta \in V\), then \(\alpha\beta \in V\). Show that if \(\alpha_0, \ldots, \alpha_4 \in V\), then there exist \begin{equation*} t_0, \ldots, t_4 \in \mathbb{Q}, \end{equation*} not all zero, such that \(t_0\alpha_0 + \ldots + t_4\alpha_4 = 0\). Deduce that if \(\alpha \in V\), then there is a polynomial, with rational coefficients, of degree at most 4 of which \(\alpha\) is a root. Find a polynomial with rational coefficients of degree 4 of which \(\alpha = \sqrt{2} + \sqrt{3}\) is a root. From this polynomial deduce the existence of an element \(\beta \in V\) such that \(\alpha\beta = 1\) and express \(\beta\) as a sum of powers of \(\alpha\). Is it true that for every non-zero \(\alpha \in V\), there is an element \(\beta \in V\) such that \(\alpha\beta = 1\)? [You may assume that \(\sqrt{2}\), \(\sqrt{3}\) and \(\sqrt{6}\) are not rational and that any set of simultaneous equations with rational coefficients in which there are more unknowns than equations has a non-zero rational solution.]
Show that if \(a\), \(b\), \(c\) are integers it is always possible to find integers \(A\), \(B\), \(C\) such that $$(a + b2^i + c2^j)(A + B2^i + C2^j) = a^2 + 2b^2 + 4c^2 - 6abc.$$ Prove that the right side of this can be zero only if the integers \(a\), \(b\), \(c\) are all zero, and deduce that if now \(a\), \(b\), \(c\) are rational numbers such that \(a + b2^i + c2^j = 0\), then \(a = b = c = 0\).
If \(p\), \(q\), \(r\), \(s\) are positive integers with \(qr - ps = 1\), prove that any fraction which lies between \(p/q\) and \(r/s\) must have denominator at least \(q + s\).
The number \(n\) whose digits in the scale of 10 are \(a\), \(b\), \(c\), \(d\) in that order is the same as the number whose digits in the scale of 9 are \(d\), \(b\), \(c\), \(a\) in that order; in other words, we have \[10^3a + 10^2b + 10c + d = 9^3d + 9^2b + 9c + a,\] and the digits \(a\), \(b\), \(c\), \(d\) all lie between 0 and 8 inclusive. Prove that there is exactly one number \(n\) (\(\neq 0\)) with this property, and find \(n\).
\(a, b, c, d\) are integers lying between 1 and 9, inclusive, and $$n = 10^4a + 10^3b + 10^2b + 10c + d.$$ $$n = b^2(10a + b)(10^2d + 10d + b)$$ is the decomposition of \(n\) into prime factors. Prove that there is exactly one \(n\) with this property, and find \(n\).
Prove that the Arithmetic Mean of a number of positive quantities is never less than their Geometric Mean. Show that if \(u+v+w=1\), and \(a \le 1\), where \(u,v,w,a\) are all positive: \[ (\frac{1}{u}-a)(\frac{1}{v}-a)(\frac{1}{w}-a) \ge 27 - 27a + 9a^2 - a^3. \]
Shew that the number of divisors (unity and the number itself included) of the number \[ N = p_1^{a_1} p_2^{a_2} \dots p_n^{a_n}, \] where \(p_1, p_2, \dots p_n\) are different primes, is \[ (a_1+1)(a_2+1)\dots(a_n+1) \equiv d(N). \] If \(f(N)\) is the number of different primes, and \(F(N)\) the total number of primes, including repetitions, in the above expression of \(N\), then \[ 2^{f(N)} \leq d(N) < 2^{F(N)}. \] In what circumstances can the signs of equality occur?
Prove that if \(a, b, c\) are in arithmetical progression, and \(a, b, d\) in harmonical progression, then \[ \frac{c}{d} = 1 - \frac{2(a-b)^2}{ab}. \]
State any rules you know for determining whether a number is divisible by 2, 3, 4, 5, 8, 9, and 11. \par Find a number of three digits, not necessarily different, such that (i) all its digits are prime, (ii) all numbers that can be formed by taking two of its digits are prime, (iii) all numbers that can be formed by taking all three of its digits are prime.
Prove that the product of any set of integers, each of which can be expressed as the sum of the squares of two integers, is equal to the sum of the squares of two integers. Express 7540 as the sum of the squares of two integers.
Prove that the arithmetic mean of \(n\) positive numbers which are not all equal exceeds their geometric mean.
\par Deduce that
\[ nx^{\frac{n-1}{2}} < 1+x+x^2+\dots+x^{n-1}, \]
so that, if \(0
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 \(a, b, c\) and \(d\) are in geometrical progression.
Find the smallest positive integer which, when divided by 28, leaves a remainder 21, and when divided by 19, leaves a remainder 17.
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 \((AB+BC+CA)abc = ABC\).
Explain the method of finding positive integral values of \(x\) and \(y\) which satisfy the equation \(ax+by=c\), where \(a, b, c\) are positive integers. Illustrate your answer by solving in positive integers the equation \(7x+11y=514\). Find the number of positive integral or zero values of \(x, y, z, w\) satisfying the equation \(x+y+2z+2w=2n\), where \(n\) is a positive integer.
The function \(\mu(n)\) is defined as being equal to 0 when \(n\) contains any squared factor, to 1 when \(n=1\), and to \((-1)^{\nu}\) when \(n=p_1 p_2 \dots p_{\nu}\), \(p_1, p_2, \dots, p_{\nu}\) being different primes. Prove that \[ \sum \mu(d)=0, \] the summation being extended to all divisors \(d\) of a given number \(N\).
A 3-inch square tile is decorated by dividing one face into 9 equal squares, and painting the resulting 1-inch squares red, green, or yellow, in such a way that no two squares with a common edge are painted the same colour. How many different tiles are possible?
A party of seven people arrives at a tavern which has six vacant rooms. In how many ways can they be accommodated if each room can take only two persons?
(i) Prove that 24 is the largest integer divisible by the product of all integers less than its square root. (ii) Show that in any set of \(n + 1\) numbers chosen from \(1, 2, ..., 2n\), there is always a pair of different numbers, one of which divides the other.
A theorem in combinatorial theory may be stated as follows: Let \(G_1, G_2, ..., G_n\) be \(n\) girls and \(B_1, B_2, ..., B_n\) \(n\) boys. In order for all boys to be able to choose as dancing partners girls with whom they are friendly, it is necessary and sufficient that, for each subset \(S\) of boys, the number of girls friendly with at least one boy in \(S\) is at least equal to the number of boys in \(S\). Prove the necessity of the condition on subsets of boys, and establish its sufficiency for \(n \leq 3\). Find the number of ways in which dancing partners may be chosen so that only friendly couples dance together, in the following situation with \(n = 5\): \begin{align*} G_1 \text{ is friendly only with } B_1, B_2, B_3 \text{ and } B_4\\ G_2 \text{ is friendly only with } B_1, B_2 \text{ and } B_5\\ G_3 \text{ is friendly only with } B_4 \text{ and } B_5\\ G_4 \text{ is friendly only with } B_1, B_2 \text{ and } B_4 \text{, and}\\ G_5 \text{ is friendly only with } B_3 \end{align*}
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 \(b\) such that \(0 < a\xi+b < 1\). Use this to show that, given any positive integer \(N\), there are distinct numbers \(n\xi+m\) and \(n'\xi+m'\) (\(n\), \(m\), \(n'\), \(m'\) all integers) whose distance apart is less than \(1/N\). Hence deduce that there are integers \(r\) and \(s\) such that \(0 < r\xi+s < 1/N\). Suppose now that \(q\) is a rational number and \(N > 0\) an integer. Prove that there is a number of the form \(n\xi + m\) between \(q\) and \(q + 1/N\). [Hint: Suppose first that \(q > 0\), and that \(r\xi+s\) has been determined as above. Consider the first integer \(k\) such that \(kr\xi + ks > q\).]
Suppose that of the 6 people at a party at least two out of every three know each other, and that all acquaintanceships are mutual. Prove that there are at least 3 people who all know each other. Does this assertion hold if the party consists of only 5 people?
Let \(x\) be a positive non-zero integer. \(S^1(x)\) will denote the sum of the digits of \(x\) when written in the scale of 10 (e.g. \(S^1(193) = 1+9+3 = 13\)). For \(i = 1, 2, \ldots\) we define \(S^{i+1}(x) = S^1(S^i(x))\). Show that \(x - S^1(x)\) is divisible by 9 for all \(i\). Denote by \(\mathcal{S}(x)\) the set \(\{S^1(x), \ldots, S^n(x)\}\) where \(n\) is the least integer such that \(S^n(x) \leq 9\) (e.g. \(\mathcal{S}(193) = \{13, 4\}\)). Show that if the smallest element of \(\mathcal{S}(x^2)\) is not a square, then it is 7, and the smallest element in \(\mathcal{S}(x)\) is 4 or 5. Deduce that if \begin{equation*} 0 < x^2 < 1000, \end{equation*} and no element of \(\mathcal{S}(x^2)\) is a square, then \(x = 4\) or 5.
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 midpoints of two opposite edges of this framework, and a beetle is placed at some point of this framework. The scorpions can move along the rods with maximum speed \(s\), and the beetle with maximum speed \(b\). Show that if \(b < 2s\) the scorpions can always catch the beetle; and explain in detail how they should manoeuvre to do so.
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 distinct primes. How many proper factors has \(N\)? (A proper factor of \(N\) is an integer \(M\) which exactly divides \(N\) and which satisfies \(1 < M < N\).) Hence or otherwise find
A spacecraft may be regarded as a solid body which is convex (i.e. no straight line meets its surface more than twice), and its total surface area is \(A\). It is required to measure a certain type of radiation. If the radiation has a certain strength, and is unidirectional (i.e. incident on the spacecraft in the form of parallel rays), the response of the detector on board is given by \(Sk\), where \(S\) is the cross-sectional area presented by the spacecraft in that direction and \(k\) is a constant. If now the spacecraft is subjected to radiation of the same total strength but isotropic (i.e. scattered equally in all directions), show that the response is \(\frac{1}{4}Ak\), whatever the shape of the spacecraft.
A finite number of circles, not intersecting or touching each other, are drawn on the surface of a sphere, thus dividing the surface into a number of regions. Prove that it is always possible to colour the surface with two colours in such a way that each region is of a single colour, and adjacent regions are of different colours. Given such a set of circles and such a colouring of the resulting regions, show that it is always possible to draw a further circle in such a way that a single recolouring of one of the new regions will restore the colour property; and that, provided there are already at least two circles present, then a further circle may be drawn in such a way that a single recolouring will not suffice.
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 number of \(i\)'s in the sequence. (Thus for \(n = 4\) we might have \(a_0, a_1, a_2, a_3 = 1, 2, 1, 0\).) If \(n \geq 7\), show that the sequence can only be $$n-4, 2, 1, 0, 0, \ldots, 0, 1, 0, 0, 0.$$ [Hint: Show that the sum of all the terms is \(n\), and that there are \(n - a_0 - 1\) non-zero terms other than \(a_0\), which sum to \(n - a_0\).]
The one-player game of Topswaps is played as follows. The player holds a pack of \(n\) cards, numbered from 1 to \(n\) in a random order. If the top card is numbered \(k\), he calls \(k\), reverses the order of the top \(k\) cards, and continues. Show that the pack eventually reaches a constant state in which the top card is numbered 1. [Hint: if \(k > 1\), and, from some point onwards, no card numbered higher than \(k\) is called, then \(k\) is called at most once thereafter.]
In a tournament everybody played against everybody else exactly once, and no game ended in a draw. Show that it is possible to order the players in such a way that everybody beat the player coming immediately after him in the ordering. Show also that if no player beat all the others then there are at least three such orderings.
The bus routes in a town have the following properties.
A convex polyhedron is such that precisely three faces concur in each vertex, and that every face is either a square or an equilateral triangle. Describe the possible cases.
5 points lie within a unit square, or on its boundary. Prove that some pair of them are at a distance apart less than or equal to \(\frac{1}{2}\sqrt{2}\), and that the smallest distance between pairs of points is only equal to \(\frac{1}{2}\sqrt{2}\) in one exceptional case.
On a chess board, which consists of 64 squares, a bishop is only allowed to move diagonally. In order to perform an '\(n\)-bounce' the Bouncing Bishop chooses a diagonal direction in which to travel and moves for \(n\) squares in such a way that on reaching the edge of the board he reverses the component of his motion perpendicular to that edge. For example, it is possible in one 6-bounce to travel from the intersection of the fourth row and fifth column to the intersection of the fourth row and third column or from the intersection of the fourth row and fourth column back to the intersection of the fourth row and fourth column. Show that, if \(n = 14k+t\) (or similarly \(14k-t\)), the effect of an \(n\)-bounce depends only on \(t\) and the initial direction of motion. Two squares are called \(n\)-equivalent if it is possible to travel from one to the other in a series of \(n\)-bounces. For each \(n\) find the number of \(n\)-equivalence classes of squares. (The \(n\)-equivalence class of a square is the set of squares which are \(n\)-equivalent to it.)
A triangle is called chromatic if all its sides are the same colour. Each pair of \(n\) distinct points \(P_1\), \ldots, \(P_n\) in space is connected with either a red line or a black line. Prove that if \(n = 6\) there must be at least 2 chromatic triangles of the form \(P_iP_jP_k\). Deduce that if \(n = 7\) there are at least 3 such chromatic triangles.
Let \(d_1, d_2, ..., d_k\) be the distinct positive divisors of the positive integer \(n\), including 1 and \(n\). Prove that \[(d_1 d_2 ... d_k)^2 = n^k.\]
Show that \(n\) coplanar lines in 'general position' (i.e. no two lines parallel, no three lines concurrent) divide the plane into \(\frac{1}{2}(n^2+n+2)\) regions. Show, also, that the regions may be coloured, each either red or blue, in such a way that no two regions whose boundaries have a line segment in common have the same colour. Find the number of regions into which \(n\) planes, in general position, divide three-dimensional space.
For any real number \(x\), \([x]\) denotes the greatest integer not exceeding \(x\). Evaluate, for positive integers \(n\), \(r\), the number of multiples of \(r\) in the set \(\{1, 2, \ldots, n\}\). Positive integers \(a_1, \ldots, a_k\) satisfy \(a_i \leq N\) (\(i = 1, \ldots, k\)) and L.C.M.\((a_i, a_j) > N\) whenever \(i \neq j\). (L.C.M.\((a_i, a_j)\) is the least common multiple of \(a_i\) and \(a_j\).) By showing that \[\sum_{i=1}^{k} [N/a_i] \leq N,\] prove that \[\sum_{i=1}^{k} \frac{1}{a_i} \leq 2.\] Deduce that, for any real \(x > 2\), \[\sum_{\sqrt{x} < p \leq x} \frac{1}{p} \leq 2,\] where the summation is over prime numbers \(p\) in the given range, and deduce that if \(r\) is a positive integer and \(N = 2^{2^r}\), then \[\sum_{1 < p \leq N} \frac{1}{p} \leq 2r.\]
A convex solid bounded by triangular faces is such that, at each vertex, either three or four edges meet. If \(x\) and \(y\) are the numbers of vertices of each type, prove that \[ 3x + 2y = 12. \] Show that, for each solution of this equation in non-negative integers, a polyhedron with the corresponding property actually exists.