Show that the equations in \(x_1, x_2, ..., x_n\) (with \(u, v\) constants): \[ux_1 x_2 + x_2 = v,\] \[ux_2 x_3 + x_3 = v,\] \[\vdots\] \[ux_{n-1} x_n + x_n = v,\] \[ux_n x_1 + x_1 = v,\] possess either one or two solutions with \(x_1 = x_2 = ... = x_n\), or else possess infinitely many solutions. Show that there are infinitely many solutions if and only if \(\sqrt{(1 + 4uv)}\) is a nonzero root of the equation \[(1+t)^n-(1-t)^n = 0,\] and hence if and only if \(uv = -\frac{1}{4} \sec^2 (\pi k/n)\) with \(1 \leq k \leq n - 1\).
Show that the composition of any two maps of the form \[z \to z_1 = \frac{az+b}{cz+d} \quad (a,b,c,d \text{ integers}; ad-bc=1)\] is of the same form and that the inverse of any map of this form is of the same form. Write down a formula for \(\operatorname{im} z_1\) involving \(\operatorname{im} z\) and show that any map of the form \(z \to z_1\) maps the upper half-plane \(H = \{z: \operatorname{im} z > 0\}\) in the complex plane onto itself. Show that any map of the form \(z \to z_1\) is a composition of maps of the form \(z \to z + n\) (\(n\) integer) and \(z \to -1/z\).
Let \(l\) be a fixed line in the plane. Let \(P\), \(Q\) be distinct points not on \(l\) lying on the same side of \(l\), and let \(\overline{P}\) be the reflexion of \(P\) in \(l\). Prove that there is a unique circle \(C\) passing through \(Q\) such that \(P\) and \(\overline{P}\) are inverse points with respect to \(C\). Define \(d(P, Q) = (\text{radius of }C)/P\overline{P}\). Prove that if \(P, Q\) are mapped to \(P', Q'\) by inversion in some circle with centre on \(l\), then \(d(P', Q') = d(P, Q)\). Deduce that \(d(Q, P) = d(P, Q)\). Let \(m\) be a line meeting \(l\). For which point \(X\) on \(m\) is \(d(P, X)\) minimum?
Let \(A\), \(B\), \(C\), \(D\) be fixed points in the plane, no three being collinear. Prove that the centres of the conics through \(A\), \(B\), \(C\), \(D\) all lie on a conic \(S\). Prove that \(S\) passes through the mid-point of \(AB\) and also through the intersection of \(AB\) and \(CD\). Show that \(S\) is a rectangular hyperbola if and only if \(A\), \(B\), \(C\), \(D\) lie on a circle.
Let \(S\) be the surface of a sphere of unit radius. The intersection of \(S\) with a plane through its centre is called a great circle. Let \(\Delta\) be a curvilinear triangle on \(S\) whose edges are arcs of great circles \(C_1, C_2, C_3\). By considering the areas of all the regions into which \(C_1, C_2, C_3\) divide \(S\), or otherwise, show that the sum of the angles of \(\Delta\) is \(\pi +\) area of \(\Delta\). A convex polyhedron with triangular faces has \(v\) vertices, \(e\) edges and \(f\) faces. Show that \(e = \frac{3f}{2}\) and \(v-e+f = 2\).
The function \(f\) satisfies the equation \[f(x) = \frac{1}{4}\left(f\left(\frac{x}{2}\right)+f\left(\frac{x+\pi}{2}\right)\right)\] for \(0 < x < \pi\). Show that if there is a constant \(M\) such that \(|f(x)| < M\) for \(0 < x < \pi\), then \(f(x) = 0\) whenever \(0 < x < \pi\). Given that \[\sum_{n=1}^{\infty} \frac{1}{(x-n\pi)^2} < 1\] for \(|x| \leq \frac{\pi}{2}\) and \(\textrm{cosec}^2 x - \frac{1}{x^2} < 1\) for \(0 < x < \frac{\pi}{2}\), prove that \[\textrm{cosec}^2 x = \sum_{n=-\infty}^{\infty} \frac{1}{(x-n\pi)^2}\] whenever \(x\) is not a multiple of \(\pi\).
The function \(f\) satisfies \(f(-y) = -f(y)\) and is defined as follows for \(y \geq 0\). \[f(y) = y \quad \text{if } 0 \leq y \leq 1,\] \[f(y) = 2-y \quad \text{if } 1 \leq y \leq 2,\] \[f(y) = 0 \quad \text{if } y \geq 2.\] Solve the differential equation \(y'' + f(y) = 0\) with initial conditions \(y(0) = 0\), \(y'(0) = c\). Sketch the solutions corresponding to initial conditions \(y(0) = 0\), \(y'(0) = c\) for \(c = 1\), \(c = \frac{4}{3}\) and \(c = \frac{8}{3}\).
A standard pack of 52 cards is thoroughly shuffled, and then dealt into four piles as follows. Cards are dealt into the first pile up to and including the first ace, then into the second pile up to and including the second ace, then into the third pile up to and including the third ace, then into the fourth pile up to and including the fourth ace, and then any remaining cards go into the first pile again. A second similar pack is thoroughly shuffled, and a single card drawn from it at random. Find the probability distribution of the size of the pile that contains the matching card from the first pack.
A population contains individuals of \(k\) types, in equal proportions. Among type \(i\), a quantity \(X\) is distributed with mean \(\mu_i\) and variance \(\sigma^2\) (the same for all \(i\)), for \(i = 1, 2, ..., k\). It is desired to estimate the mean of \(X\) over the whole population. Two methods of estimation are considered. In the first a random sample of size \(n\) (with replacement) is drawn from each of the \(k\) types, and in the second a random sample of size \(kn\) is drawn (with replacement) from the whole population without regard to type. In each case the mean of the \(kn\) \(X\)-values is computed. Show that the expectation of the resulting estimate is in each case \[\mu = \frac{1}{k}\sum_{i=1}^k \mu_i,\] but that the second estimate has variance greater than that of the first by an amount \[\frac{1}{k^2n}\sum_{i=1}^k (\mu_i-\mu)^2.\]
Solution: Let \(X_{i,j} \sim N(\mu_i, \sigma^2)\) be iid. Let \(Y_i\) be a sample from the second distribution, In the first case: \begin{align*} && \mathbb{E}\left ( \frac{1}{kn} \sum_{i=1}^k \sum_{j=1}^n X_{i,j} \right) &=\frac{1}{kn} \sum_{i=1}^k \sum_{j=1}^n\mathbb{E}\left ( X_{i,j} \right) \\ &&&= \frac{1}{kn} \sum_{i=1}^k \sum_{j=1}^n\mu_i \\ &&&= \frac{1}{kn} \sum_{i=1}^k n\mu_i \\ &&&= \frac{1}{k}\sum_{i=1}^k \mu_i \end{align*} In the second case: \begin{align*} && \mathbb{E}\left ( \frac{1}{kn} \sum_{i=1}^{kn} Y_i \right) &=\frac{1}{kn} \sum_{i=1}^{kn} \mathbb{E} \left (Y_i \right) \\ &&&=\frac{1}{kn} \sum_{i=1}^{kn} \mathbb{E} \left (Y_i |Y_i \text{ is of type }T\right) \\ &&&=\frac{1}{kn} \sum_{i=1}^{kn} \mathbb{E} \left (\mu_T\right) \\ &&&= \frac{1}{kn} \sum_{i=1}^{kn} \frac1k \left ( \sum_{j=1}^{k}\mu_j\right) \\ &&&= \frac1k \sum_{j=1}^{k}\mu_j \\ \end{align*} so they are equal. \begin{align*} && \textrm{Var}\left ( \frac{1}{kn} \sum_{i=1}^k \sum_{j=1}^n X_{i,j} \right) &= \frac{1}{k^2n^2} \sum_{i=1}^k \sum_{j=1}^n \textrm{Var}\left ( X_{i,j} \right) \\ &&&= \frac{1}{k^2n^2} \sum_{i=1}^k \sum_{j=1}^n \sigma^2 \\ &&&= \frac{\sigma^2}{kn} \end{align*} \begin{align*} && \textrm{Var}\left ( \frac{1}{kn} \sum_{i=1}^{kn} \sum_{j=1}^n Y_i \right) &= \frac{1}{k^2n^2}\sum_{i=1}^{kn} \textrm{Var}\left ( Y_i \right) \\ &&&= \frac{1}{k^2n^2}\sum_{i=1}^{kn} \mathbb{E}\left (Y_i^2 \right)-\frac1{kn}\mu^2 \\ &&&= \frac{1}{k^2n^2}\sum_{i=1}^{kn} \mathbb{E} \left ( \mathbb{E}\left (Y_i^2 | Y_i \text{ is of type }T \right)\right)-\frac1{kn}\mu^2 \\ &&&= \frac{1}{kn}\sum_{j=1}^{k} \frac{1}{k} \left (\sigma^2+\mu_j^2 \right)-\frac1{kn}\mu^2 \\ &&&= \frac{1}{kn}\sum_{j=1}^{k} \frac{1}{k} \left (\sigma^2+\mu_j^2 \right)-\frac1{kn}\mu^2 \\ &&&= \frac{\sigma^2}{kn}+\frac{1}{k^2n}\sum_{j=1}^k\mu_j^2-\frac1{kn}\mu^2 \\ &&&= \frac{\sigma^2}{kn}-\frac{1}{k^2n}\sum_{j=1}^k(\mu^2-\mu_j^2) \\ \end{align*} \begin{align*} \sum_{i=1}^k (\mu_i-\mu)^2 &= \sum_{i=1}^k \mu_i^2-2\sum_{i=1}^k\mu_i\mu+k\mu^2 \\ &= \sum_{i=1}^k \mu_i^2-2k\mu^2+k\mu^2 \\ &= \sum_{i=1}^k \mu_i^2-k\mu^2 \\ &= \sum_{i=1}^k \left(\mu_i^2-\mu^2\right) \end{align*} as required.
Let \((r, \theta)\) be polar coordinates in the plane of a lamina. If \(I(\theta)\) is the moment of inertia of the lamina about the line \(\theta =\) constant, show that there exist constants \(A\), \(B\) and \(\alpha\) such that \[I(\theta) = A + B\sin(2\theta+\alpha).\] Deduce that if the lamina has the same moment of inertia about three different lines in its plane, all of which pass through a fixed point \(O\), then it has the same moment of inertia about every line in its plane through \(O\). A uniform lamina of mass \(M\) has the shape of a regular \(n\)-sided polygon inscribed in a circle of radius \(a\). Calculate its moment of inertia about an edge.