Prove that, if \(a\) and \(b\) are integers, then \(6a + 5b\) is divisible by 13 if and only if \(3a - 4b\) is. Determine all positive integers \(k\) such that if \(a\) and \(b\) are integers then \(6a + 5b\) is divisible by \(k\) if and only if \(3a - 4b\) is.
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 \(n\) be an integer and let \(p\) be a prime. Prove that the exponent of \(p\) in the prime factorization of \(n!\) is given by \(\frac{n-s}{p-1}\), where \(s\) is the sum of the digits of \(n\) when written to the base \(p\). How many zeros are at the end of 1000!, when written to the base 60? [You are reminded that every integer \(n\) can be written as \(n = p_1^{a_1} \cdot \ldots \cdot p_k^{a_k}\), where \(p_1 < p_2 < \ldots < p_k\) are primes and \(a_1, \ldots, a_k\) are integers. The exponent of \(p\) in the prime factorization of \(n\) is \(a_i\) if \(p = p_i\) for some \(i\), \(1 \leq i \leq k\); otherwise it is zero.]
\(ABCDE\) is a regular pentagon inscribed in a circle, and \(A'\) is the other extremity of the diameter through \(A\). Prove that \[DC^2 - A'C^2 = \frac{1}{4}A'A^2.\]
A circle touches the ellipse \(x^2/a^2 + y^2/b^2 = 1\) at its intersections with the line \(x = c\). Find its centre and radius. Interpret your results when \(c\) is formally put equal to (i) \(a\), (ii) a value strictly between \(a\) and \(a/e\), (iii) \(a/e\), where \(e\) is the eccentricity of the ellipse.
The Cartesian coordinates of a particle \(P\) at time \(t\) are \((x(t), y(t))\), where \[x = u(1+t), \quad (u > 0),\] \[\frac{dy}{dx} = \frac{y}{x} + \frac{x}{(x^2+y^2)^{\frac{1}{2}}}.\] Initially the particle is on the \(x\) axis; if \(O\) is the origin \((0, 0)\), prove that the slope of \(OP\) increases with time, and show that \(4y = 3x\) after a time \[t = \sqrt{2}\exp(15/32) - 1.\]
By applying the Taylor expansion to the function \(f(x) \equiv (x^2-1)^n\), or otherwise, prove that for all \(x\), and \(h \neq 0\), \[\left[\frac{(x^2-1) + 2hx + h^2}{h}\right]^n = \sum_{r=0}^{2n} \frac{h^{r-n}}{r!}\left(\frac{d}{dx}\right)^r[(x^2-1)^n].\] Write \((x^2-1)/h\) for \(h\) on each side of the above equation, and show that for \(1 \leq m \leq n\), \[\frac{1}{(n-m)!}\left(\frac{d}{dx}\right)^{n-m}[(x^2-1)^n] = \frac{1}{(n+m)!}(x^2-1)^m\left(\frac{d}{dx}\right)^{n+m}[(x^2-1)^n].\] Deduce that \[y(x) = \left(\frac{d}{dx}\right)^n[(x^2-1)^n]\] satisfies the differential equation \[\frac{d}{dx}\left[(x^2-1)\frac{dy}{dx}\right] - n(n+1)y = 0.\]
For each integer \(n \geq 1\), write \(t_n\) for the number of ways of placing \(n\) people into groups (so that \(t_1 = 1\), \(t_2 = 2\), \(t_3 = 5\), etc.). Defining \(t_0 = 1\), show that \[t_{n+1} = \sum_{r=0}^{n} \binom{n}{r}t_{n-r},\] for \(n \geq 0\), and hence show that \(t_n/n!\) is the coefficient of \(x^n\) in the Maclaurin expansion of \(\exp(\exp x - 1)\), for each \(n \geq 1\).
In an election there are three candidates, \(A, B\) and \(C\), and \(N\) voters. Each voter acts independently of the others, and is equally likely to vote for any one of the candidates. Each voter votes exactly once. Suppose the voters are numbered \(1, 2, \ldots, N\) and define the random variables \(A_1, \ldots, A_N, B_1, \ldots, B_N\) by \begin{align*} A_j &= \begin{cases} 1 & \text{if the \(j\)th voter votes for \(A\),} \\ 0 & \text{otherwise}; \end{cases} \\ B_j &= \begin{cases} 1 & \text{if the \(j\)th voter votes for \(B\),} \\ 0 & \text{otherwise}. \end{cases} \end{align*} Let \(X, Y\) be the total number of votes cast for \(A, B\) respectively. Find the distribution of \(X\), and the covariance of \(X\) and \(Y\). Why would you expect this covariance to be negative?
(i) The real numbers \(a_1, \ldots, a_n\) satisfy the constraint \begin{equation*} \sum_{i=1}^{n} a_i = C, \end{equation*} where \(C\) is a given constant. Show that \(\sum_{i=1}^{n} a_i^2\) is minimised subject to (*) by \(a_i = C/n\) for \(i = 1, \ldots, n\). (ii) In an experiment to determine the mean body-weight \(\mu\) of a species of moth, \(n\) moths of this species are weighed, and their weights \(x_1, \ldots, x_n\) recorded. It may be assumed that \(x_1,\ldots, x_n\) are uncorrelated and have common mean \(\mu\) and common variance \(\sigma^2\), where \(\sigma^2\) is known. We wish to find the best linear unbiased estimator of \(\mu\), that is the function \(\sum_{i=1}^{n} a_i x_i\) which has expectation \(\mu\) and smallest variance. Assuming (i), find the appropriate values of the set \(\{a_i\}\), and find the variance of the best linear unbiased estimator.