Show that given an arithmetic progression \(a_n\) of integers, if one of the members is the cube of an integer then so are infinitely many others. Show also that if \(a_n = 7n + 5\) then no \(a_n\) is the cube of an integer.
Let \(A\), \(B\) be real \(2 \times 2\) matrices. Show that only one of the following assertions is always true by proving it and supplying counterexamples to the others:
The Parliament of the democratic state of Steinmark has \(r\) members. Much business is conducted not by the full House but by a system of Standing Committees, of which there are \(N\) altogether; the Constitution decrees that each committee shall have exactly \(q\) members, and each member of Parliament shall belong to exactly \(n\) committees. A political commentator observes the remarkable fact that there is an integer \(p < r\) such that, for each set of \(p\) members of Parliament, there is exactly one committee to which they all belong. Find expressions in terms of \(p\), \(q\) and \(r\) for \(N\) and \(n\). Suppose now that \(p = 4\) and \(q = 9\). By considering the number of committees to which three given members all belong, show that \(r - 3\) is divisible by 6. Hence show that either \(r - 1\) or \(r - 3\) is divisible by 8, and that either \(r\) or \(r - 3\) is divisible by 9.
Describe how to construct a right-angled triangle \(ABC\) (with the right angle at \(C\)) given the lengths \(t_a\), \(t_b\) of the medians from \(A\) to the midpoint of \(BC\) and from \(B\) to the midpoint of \(AC\), respectively. Prove also that a necessary and sufficient condition for such a triangle to exist is that \[t_a < 2t_b \quad \text{and} \quad t_b < 2t_a.\] [Hint: Consider a suitable parallelogram. You may assume a construction enabling you to find one third of a given length.]
Let \([x]\) denote the integer part of \(x\). Sketch the graph of \(\left[1 + \frac{x}{\pi}\right]\) for \(x \geq 0\). Find the solution \(y(x)\) for \(x \geq 0\) of the differential equation \[\frac{d^2y}{dx^2} + \left[1 + \frac{x}{\pi}\right]^2 y = 0,\] subject to the conditions \(y = 0\) and \(dy/dx = 1\) at \(x = 0\). You may assume that \(y\) and \(dy/dx\) are continuous everywhere. Show that \[\int_0^{\infty} y^2dx = \frac{\pi}{2} \sum_{n=1}^{\infty} \frac{1}{n^2}.\]
At time \(t = 0\), 4 insects \(A\), \(B\), \(C\) and \(D\) stand at the corners of a square of side \(a\). For time \(t > 0\) each insect crawls with constant speed \(v\) in the direction of the next insect in cyclic order (that is, \(A\) crawls towards \(B\), \(B\) towards \(C\), and so on). Show that the insects meet after a time \(a/v\) and that they encircle the centre of the square an infinite number of times. [Hint: Use polar coordinates.]
A circle of radius \(a\) rolls without slipping around the outside of a circle of radius \(2a\). Show that the arc length of a curve traced out by a point on its circumference is \(24a\).
Let \(x_n\) be the \(n\)th positive root of the equation \[ax = \tan x, \quad a > 0.\] (i) Show that, for small \(a\), \(x_1 \approx \pi(1 + a + a^2)\). (ii) Show that, for large \(n\), \(x_n \approx (n + \frac{1}{2})\pi - \frac{1}{a(n + \frac{1}{2})\pi}\) if \(a < 1\); what is the corresponding result for \(a > 1\)?
Independent random variables \(X_1, \ldots, X_n\) have a uniform distribution on the interval \([\theta-\frac{1}{2}, \theta+\frac{1}{2}]\), where \(\theta\) is unknown. Let \[V = \max\{X_1, \ldots, X_n\}, \quad U = \frac{1}{n}\sum_{i=1}^{n} X_i.\] Show that if \(\theta = \theta_0\), \[P(V \leq x) = \left (x-\theta_0+\frac{1}{2} \right)^n \quad \text{for } \theta_0-\frac{1}{2} \leq x \leq \theta_0+\frac{1}{2}.\] Calculate the mean and variance of \(V\) and \(U\). If \(W = V - c\), where \(c\) is chosen so that \(EW = \theta_0\), compare \(\text{var}\ U\) and \(\text{var}\ W\). Suppose that \(n\) is large. Which of \(U\), \(W\) would you use if you wished to estimate \(\theta\), and why?
Solution: \begin{align*} P(V \leq x) &= P(\max(X_1, \ldots, X_n) \leq x) \\ &= P(X_1 \leq x) \cdots P(X_n \leq x) \\ &= \frac{\left ( x - \left ( \theta_0 - \frac12\right) \right)}{1} \cdots \frac{\left ( x - \left ( \theta_0 - \frac12\right) \right)}{1} \\ &= \left ( x - \left ( \theta_0 - \frac12\right) \right)^n \end{align*} \begin{align*} \mathbb{E}(U) &= \mathbb{E} \left ( \frac{1}{n}\sum_{i=1}^{n} X_i\right) \\ &= \frac{1}{n}\sum_{i=1}^{n} \mathbb{E} \left ( X_i\right) \\ &=\frac{1}{n}\sum_{i=1}^{n}\theta_0 \\ &= \theta_0 \\ \textrm{Var}(U) &= \textrm{Var}\left ( \frac{1}{n}\sum_{i=1}^{n} X_i\right) \\ \\ &= \frac1{n^2} \sum_{i=1}^{n} \textrm{Var}\left ( X_i\right) \\ &= \frac{1}{n^2} \frac{n}{12} \\ &= \frac{1}{12n} \\ \\ \mathbb{E}(V) &= \int_{\theta_0-\frac12}^{\theta_0+\frac12}xn \left ( x - \left ( \theta_0 - \frac12\right) \right)^{n-1} \d x \\ &= n\int_0^1 \left (u + \left ( \theta_0 - \frac12\right) \right)u^{n-1} \d u \\ &= n \left [ \frac{u^{n+1}}{n+1} + \left ( \theta_0 - \frac12\right) \frac{u^{n}}{n}\right]_0^1 \\ &= \frac{n}{n+1} + \theta_0 - \frac12 \\ \textrm{Var}(V) &= \mathbb{E}(V^2) - \left [ \mathbb{E}(V) \right]^2 \\ &= \int_{\theta_0-\frac12}^{\theta_0+\frac12}x^2n \left ( x - \left ( \theta_0 - \frac12\right) \right)^{n-1} \d x - \left (\frac{n}{n+1} + \theta_0 - \frac12 \right)^2 \\ &= n\int_0^1 \left (u + \theta_0 - \frac12 \right)^2u^{n-1} \d u - \left (\frac{n}{n+1} + \theta_0 - \frac12 \right)^2 \\ &= n \left [ \frac{u^{n+2}}{n+2} + (2 \theta_0 - 1) \frac{u^{n+1}}{n+1} + \left ( \theta_0 - \frac12 \right)^2 \frac{u^n}{n}\right]_0^1 - \left (\frac{n}{n+1} + \theta_0 - \frac12 \right)^2 \\ &= \frac{n}{n+2} + \frac{n}{n+1} \left ( 2\theta_0 - 1 \right) + \left ( \theta_0 - \frac12 \right)^2 - \left (\frac{n}{n+1} + \theta_0 - \frac12 \right)^2 \\ &= \frac{n}{n+2} - \frac{n^2}{(n+1)^2} \\ &= \frac{n(n+1)^2-n^2(n+2)}{(n+2)(n+1)^2} \\ &= \frac{n^3+2n^2+n - n^3-2n^2}{(n+2)(n+1)^2} \\ &= \frac{n}{(n+2)(n+1)^2} \end{align*} Therefore the variance of \(V\) is much smaller than \(U\) for large \(n\), so we should choose that.
The Royal Mint wishes to determine whether a given coin is fair or not, and has decided to conduct the following test. The coin will be tossed \(N\) times, and will be deemed fair if \(X\), the proportion of heads, lies in the interval \((a, b)\). Let \(p\) be the (unknown) probability of obtaining heads on a single toss of the coin. \(N\), \(a\), \(b\) must satisfy the conditions: (i) If \(p = 0.5\), \(P(a \leq X \leq b) \geq 0.95\). (ii) If \(p \geq 0.51\), \(P(X > b) \geq 0.95\). (iii) If \(p \leq 0.49\), \(P(X < a) \geq 0.95\). Find the approximate value of the smallest possible value of \(N\), and appropriate values of \(a\) and \(b\) for this \(N\). The experiment was conducted with \(N = 40,000\). State what conclusions may be drawn from (a) 20,400 heads, (b) 19,900 heads.