Problems

1. [(i)] If \(a_1, a_2, a_3\) are the roots of \[ x^3+px+q=0, \] where \(p+q+1 \ne 0\), find the equation with roots \[ 1/(1-a_i) \quad (i=1, 2, 3). \]
2. [(ii)] Solve the equations \begin{align*} x+y+z &= 7, \\ x^2+y^2+z^2-4z &= 5, \\ xyz-2xy &= 4. \end{align*}

If \(x_1, \dots, x_n\) are real numbers, prove that \[ n(x_1^2+\dots+x_n^2) \ge (x_1+\dots+x_n)^2 \] and determine when equality occurs. Prove that \[ \sum_{r=1}^n \frac{1}{r^2+r} \le \frac{n}{(n+1)}. \]

A sequence \(u_0, u_1, \dots\) is defined by \(u_0=3\), \(u_{n+1}=(2u_n+4)/u_n\). Prove that

1. [(i)] \(3 \le u_n \le \frac{10}{3}\) for all \(n\);
2. [(ii)] \(|u_n-a| \le (\frac{2}{3})^n (a-3)\) for all \(n\), where \(a\) is the positive root of \(x^2-2x-4=0\).

Two polynomials \(f_0(x), f_1(x)\) are given and a sequence of polynomials \(f_2(x), f_3(x), \dots, f_r(x)\) are defined by the rule that \(f_{i+1}(x)\) is the remainder when \(f_{i-1}(x)\) is divided by \(f_i(x)\), \(f_r(x)\) being the last such remainder that is different from zero. Prove, by induction on \(i\), that

1. [(i)] each of the polynomials \(f_i(x)\) can be expressed in the form \(a_i(x)f_0(x)+b_i(x)f_1(x)\), where \(a_i(x)\) and \(b_i(x)\) are polynomials;
2. [(ii)] \(f_r(x)\) divides \(f_i(x)\) for \(i=1, \dots, r\).

If \(f_r(x)\) is a constant and \(f_0(x)\) divides \(f_1(x)g(x)\), prove that \(f_0(x)\) divides \(g(x)\).

If \(u_0 = \sinh\alpha\), \(u_1=\sinh(\alpha+\beta)\) and \(u_{n+2}-2u_{n+1}\cosh\beta+u_n=0\) for all \(n \ge 0\), prove that \(u_n = \sinh(\alpha+n\beta)\). Sum the series \[ \sum_{r=0}^n \sinh(\alpha+r\beta) \] for all values of \(\beta\).

Find

1. [(i)] \(\int_0^1 \cos^{-1}\sqrt{1-x^2} dx\),
2. [(ii)] \(\int_0^1 \frac{dx}{1+x^2+x^4}\),
3. [(iii)] \(\int \frac{dx}{(x^2+1)^{\frac{1}{2}}+(x^2-1)^{\frac{1}{2}}}\).

If \(y=\sin^{-1}x\), show that \(y''(1-x^2)=xy'\). By Leibniz' Theorem or otherwise, find \(y^{(n)}\) at \(x=0\) and hence expand \(\sin^{-1}x\) as a power series in \(x\).

Sketch the curve whose equation, in Cartesian coordinates, is \[ x^4 - 2xy^2 + y^4 = 0. \]

If \[ I_{m,n} = \int_0^\pi \sin^m x \sin nx dx, \] obtain a relation between \(I_{m,n}\) and \(I_{m-2,n}\) for \(m \ge 2\). Hence evaluate \(I_{4,5}\).

In a heat of a certain beauty contest, there are six girls competing and two judges. Each judge lists the girls in order of merit, and for a girl to go forward to the final she must be one of the first three on each list. Assuming that the lists are independent and bear no relation to the charms of the competitors, what is the chance that just two girls will go on to the final?