Problems

Sydney Smith (1771--1845), clergyman and celebrated wit, once comforted a friend with these words: `The cholera will have killed by the end of the year about one person in every thousand. Therefore it is a thousand to one (supposing the cholera to travel at the same rate) that any person does not die of the cholera in any one year. This calculation is for the mass; but if you are prudent, temperate and rich, your chance is at least five times as good that you do not die of cholera in a year; it is not far from two millions to one that you do not die any one day from cholera. It is only seven hundred and thirty thousand to one that your house is not burnt down any one day. Therefore it is nearly three times as likely that your house should be burnt down any one day, as that you should die of cholera; or, it is as probable that your house should be burnt down three times in any one year, as that you should die of cholera.' Expose the major fallacy in this argument in language that Sydney Smith would have understood.

Discuss the solution of the system of linear equations \begin{align} x + y + z + t &= 4,\\ x + 2y + 3z + 4t &= 10,\\ x + 4y + az + bt &= c, \end{align} with due regard to the special cases which may arise for particular values of \(a\), \(b\) and \(c\).

1. Solve $$x^4 - x^3 - 4x^2 - x + 1 = 0.$$
2. Solve $$2^x + 8^x = 4^{x+1}.$$
3. Find the positive root of $$\log_4(x + 1) = \frac{3}{4}(\log_2 x + \log_3 x).$$

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.

\(x_1, \ldots, x_n\) are distinct numbers and, for \(1 \leq r \leq n\), \(p_r(x)\) is written for $$(x - x_1) \ldots (x - x_{r-1})(x - x_{r+1}) \ldots (x - x_n).$$ By considering $$\sum_{r=1}^{n} \alpha_r p_r(x),$$ for suitably chosen \(\alpha_r\), show that it is possible to find a polynomial of degree not exceeding \(n-1\) which takes given values at \(x_1, \ldots, x_n\). Similarly, by considering $$\sum_{r=1}^{n} (\beta_r x + \gamma_r)\{p_r(x)\}^3,$$ show that it is possible to find a polynomial of degree not exceeding \(2n-1\) which takes given values at \(x_1, \ldots, x_n\) and whose first derivative also takes given values at these points.

1. [(i)] By considering the series expansion of \(e^{-x}(e^x - 1)^{n+1}\), or otherwise, show that $$n^n - (n+1)(n-1)^n + \frac{(n+1)n}{2!}(n-2)^n - \ldots(\text{to } n \text{ terms}) = 1.$$
2. [(ii)] Show that $$1^n - n2^n + \frac{n(n-1)}{2!}3^n - \ldots(\text{to } n+1 \text{ terms}) = (-1)^n n!.$$

1. [(i)] Prove, by induction or otherwise, that \(3^{2n+1} + 2^{n+2}\) is divisible by 7 for any positive integer \(n\).
2. [(ii)] Let $$u(m, n) = \{(n+1)!\}^2 + (n-1)!^{3m} + \{(n+1)! - (n-1)!\}^{2m},$$ where \(m\) and \(n\) are positive integers. Show that, for each pair of values of \(m\) and \(n\), \(u(m, n)\) is an integer. Show also (by finding a relation between \(u(m, n)\), \(u(m+1, n)\) and \(u(m+2, n)\), or otherwise) that \(u(m, n)\) is divisible by \(2^{m+1}\).

\(C_1\) and \(C_2\) are two circles; the polars of a point \(A\) with respect to \(C_1\) and \(C_2\) meet at \(B\). Prove that \(B\) is on the radical axis of \(C_1\) and \(C_2\) if and only if \(A\) is also.

A point moves so that its least distances from each of two fixed circles are equal; describe its locus in each of the various cases that may arise, and justify your statements.

Three tangents are drawn to a parabola so that the sum of the angles which they make with the axis of the parabola is \(\pi\). Prove that the circumcircle of the triangle formed by the tangents touches the axis of the parabola at the focus.