Problems

Show that, if \(p \neq 0\) and \(4p^3 + 27q^2 \neq 0\), the cubic polynomial \(x^3 + px + q\) can be expressed in the form \[ \frac{\alpha(x - \beta)^3 - \beta(x - \alpha)^3}{\alpha - \beta} \] where \(\alpha\) and \(\beta\) are certain constants. Hence, or otherwise, prove that, if \(4p^3 + 27q^2 \neq 0\), the cubic equation \(x^3 + px + q = 0\) has three unequal roots, which can be found by solving a quadratic equation and a cubic equation of the special type \(x^3 = a\).

A given conic has equation \(S=0\); the tangent at a fixed point \(P\) of the conic has equation \(t=0\). Write down equations representing the most general conics satisfying the following conditions:

1. [(i)] touching the given conic at \(P\),
2. [(ii)] having three-point contact with the given conic at \(P\).

A fixed conic \(S\) touches the sides \(AB, AC\) of a triangle at \(B\) and \(C\). A conic \(S'\) has three-point contact with \(S\) at \(B\) and passes through \(C\). Prove that (i) if the tangent to \(S'\) at \(C\) cuts \(AB\) in \(D\), and \(E\) is the harmonic conjugate of \(B\) with respect to \(A\) and \(D\), then \(E\) lies on the common tangent of \(S\) and \(S'\) other than \(AB\); (ii) the centres of all conics \(S'\) lie on a fixed conic touching \(AB\) at \(B\).

A segment of height \(\frac{1}{4}a\) is cut off by a plane from a uniform solid sphere of radius \(a\). If this segment can rest with its curved surface in contact with an inclined plane that is rough enough to prevent slipping, show that the angle between the plane and the horizontal cannot exceed \(\sin^{-1}(27/40)\).

Two points \(P\) and \(Q\) are inverse with respect to a circle \(\Sigma\). The inverses of \(\Sigma, P, Q\) with respect to another circle \(\Gamma\) are \(\Sigma', P', Q'\). Prove that \(P'\) and \(Q'\) are inverse with respect to \(\Sigma'\). Prove that the locus of the inverse points of a given point with respect to a system of coaxal circles is a circle which cuts the system orthogonally.

If \[ D_n = \begin{vmatrix} a & b & 0 & 0 & \dots & 0 & 0 \\ c & a & b & 0 & \dots & 0 & 0 \\ 0 & c & a & b & \dots & 0 & 0 \\ 0 & 0 & c & a & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \dots & a & b \\ 0 & 0 & 0 & 0 & \dots & c & a \end{vmatrix}, \] where the determinant is of order \(n\), obtain a relation connecting \(D_n\), \(D_{n-1}\) and \(D_{n-2}\). Hence, or otherwise, prove (i) that, if \(a=2, b=c=1\), then \(D_n = n+1\); and (ii) that if \(a=b=c=1\), then \(D_n=1, 1, 0, -1, -1, 0\), according as \(n\) leaves the remainder \(0, 1, 2, 3, 4, 5\), when divided by 6.

Obtain formulae for the number of permutations \({}^n P_r\) and the number of combinations \({}^n C_r\) of \(n\) unlike things \(r\) at a time. There are \(n\) cards numbered from 1 to \(n\) in black and from \(n\) to 1 in red so that the two numbers on each card add up to \(n+1\). Prove that the number of groups of \(r\) cards which can be formed so that the \(2r\) numbers on them are all different is \[ \frac{n(n-2)(n-4)\dots(n-2r+2)}{r!} \] if \(n\) is even, and find the corresponding formula when \(n\) is odd. Hence, or otherwise, find the probability that, when four of the numbers \(1, 2, 3, 4, 5, 6, 7, 8\) are taken at random, no two of them add up to 9.

A weight \(W\) is attached to a fixed point by four light strings. At the mid-point of each string, which is of length \(2l\), is fixed a light particle. Any particle repels any other with a force equal to \(\lambda\) times the reciprocal of the square of their distance apart. Show that, if the particles are in equilibrium at the vertices of a square of side \(x\), \[ W^2 x^6 - (9+4\sqrt{2})\lambda x^2 (2l^2 - x^2) = 0. \] Prove that this equation has one and only one positive root.

A point \(P\) moves along a fixed line and \(O\) is a fixed point not on the line; find the envelope of the line through \(P\) perpendicular to \(OP\). A variable ellipse has a given focus and touches two given lines; prove that the envelope of its minor axis is a parabola.

By induction, or otherwise, prove the identity \[ \frac{(1-x^{n+1})(1-x^{n+2})(1-x^{n+3})\dots(1-x^{2n})}{(1-x)(1-x^3)(1-x^5)\dots(1-x^{2n-1})} = (1+x)(1+x^2)(1+x^3)\dots(1+x^n), \] where \(n\) is any positive integer. Prove that the expansions of \[ (1+x)(1+x^2)(1+x^3)\dots(1+x^n) \] and \[ (1-x)^{-1}(1-x^3)^{-1}(1-x^5)^{-1}\dots(1-x^{2n-1})^{-1} \] in ascending powers of \(x\) agree as far as the terms in \(x^n\). Hence, or otherwise, prove that the number of ways of expressing a positive integer \(n\) as a sum of one or more unequal positive integers is the same as the number of ways of expressing \(n\) as a sum of one or more odd (not necessarily unequal) positive integers.

Prove de Moivre's theorem \((\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta\) for any integer \(n\) (positive, negative, or zero). Explain how this result can be used to deduce trigonometrical identities from algebraical identities, and illustrate by proving the identity \[ \frac{2.4.\dots.2n}{1.3.\dots.(2n-1)}\cos^{2n}\theta = 1 + 2\sum_{r=1}^n \frac{n(n-1)\dots(n-r+1)}{(n+1)(n+2)\dots(n+r)}\cos 2r\theta, \] where \(n\) is any positive integer.