Problems

The sides \(BC, CA, AB\) of a triangle \(ABC\) are cut by a straight line in \(D, E, F\) respectively. Prove that \[ AF \cdot BD \cdot CE = AE \cdot BF \cdot CD. \] A circle concentric with the circumcircle of the triangle \(ABC\) cuts \(AC\) in \(Q, Y\) and cuts \(AB\) in \(R, Z\). \(QR, YZ\) cut \(BC\) in \(M\) and \(N\). Prove that \(M\) and \(N\) are equidistant from the centre of the circle.

Prove that the foot of the perpendicular from the focus of a parabola on any tangent lies on the tangent at the vertex. A straight line cuts a circle; prove that all the chords of the circle that are bisected by the straight line touch a parabola.

\(S\) and \(H\) are the foci of an ellipse. \(P\) and \(Q\) are the points of contact of the tangents from \(T\). Prove that a circle can be drawn with centre \(T\) to touch \(SP, HP, SQ\) and \(HQ\), produced if necessary.

\(ABCD\) is a tetrahedron. By drawing pairs of parallel planes through the pairs of opposite edges a parallelepiped is described. Prove that its volume is three times that of the tetrahedron. If \(AB\) is perpendicular to \(CD\), and if \(AC\) is perpendicular to \(BD\), prove that \(AD\) is perpendicular to \(BC\).

The tangent at any point \(P\) of the parabola \(y^2=4ax\) is met in \(Q\) by a line through the vertex \(A\) at right angles to \(AP\), and \(Z\) is the foot of the perpendicular from \(A\) on the tangent at \(P\). Show that there are three positions of the point \(P\) on the parabola for which \(Z\) lies on the straight line \(lx+my+na=0\), and that the corresponding points \(Q\) lie on the line \[ (2l-n)x + 4my + 2na = 0. \]

Prove that the common chords of a circle and an ellipse are equally inclined to the axes of the ellipse. A circle passes through the focus \((ae,0)\) of the ellipse \(x^2/a^2+y^2/b^2=1\), and touches the ellipse at the point whose eccentric angle is \(\phi\). Prove that the line joining the other two points of intersection of the two curves is \[ (x/a - 1/e) \cos\phi - (y/b) \sin\phi = (1-e^2)/e^2. \]

1. [(i)] Solve the equation \[ \frac{4}{x^2-2x} - \frac{2}{x^2-x} = x^2-x. \]
2. [(ii)] Prove that \(x = \dfrac{2}{n+1}\) satisfies the equation \[ (x-1)^3+(2x-1)^3+(3x-1)^3+\dots+(nx-1)^3=0, \] and find the quadratic equation satisfied by the other two roots.

1. [(i)] Sum to \(n\) terms the series \[ \frac{r}{r+1!} + \frac{2r^2}{r+2!} + \frac{3r^3}{r+3!} + \dots. \]
2. [(ii)] Sum to infinity the series \[ 1 + \frac{11}{14} + \frac{11 \cdot 13}{14 \cdot 16} + \frac{11 \cdot 13 \cdot 15}{14 \cdot 16 \cdot 18} + \dots \] and establish its convergence.

1. [(i)] Find the condition that two roots of the equation \[ x^3+3px+q=0 \] should be equal.
2. [(ii)] If the roots of the equation \[ x^3+lx^2+mx+n=0 \] are the cosines of the angles of a triangle, prove that \[ l^2=2m+2n+1. \]

By considering the expansions of \((e^x-1)^n\) and of \(\dfrac{1}{1-x+cx^2}\) or otherwise, prove that, if \(n\) is a positive integer,

1. [(i)] \(n^{n+1} - n(n-1)^{n+1} + \frac{n(n-1)}{2!}(n-2)^{n+1} - \dots = \frac{n}{2}(n+1)!\),
2. [(ii)] \(1-(n-1)c + \frac{(n-2)(n-3)}{2!}c^2 - \frac{(n-3)(n-4)(n-5)}{3!}c^3+\dots = \frac{m^{n+1}-1}{(m-1)(m+1)^n}\),