Problems

Let \(n\) be a non-negative integer. Show that the number of solutions of \[x + 2y + 3z = 6n\] in non-negative integers \(x\), \(y\) and \(z\) is \(3n^2 + 3n + 1\). Find the corresponding number for the equation \[x + 2y + 3z = 6n + 1.\]

Show that the square of any odd integer is congruent to 1 modulo 8. Let \(R\) be the ring of integers taken modulo 8 and let \(G\) be the group of all \(2 \times 2\) matrices \[\begin{pmatrix} u & x \\ y & v \end{pmatrix}\] with entries in \(R\) such that \(u\) and \(v\) are both odd, \(x\) and \(y\) are both even and \(uv - xy = 1\). Determine the number of elements in \(G\) and the number of elements of order 2 in \(G\). [You need not verify that \(G\) is a group.]

The cubic equation \[x^3 + 3qx + r = 0 \quad (r \neq 0)\] has roots \(\alpha\), \(\beta\) and \(\gamma\). Verify that the sextic equation \[r^2(x^2 + x + 1)^3 + 27q^3x^2(x + 1)^2 = 0\] is satisfied by \(\alpha/\beta\). Comment on this result in relation to the roots of the cubic in the cases (i) \(q = 0\) and (ii) \(4q^3 + r^2 = 0\).

Let \(d_1, d_2, ..., d_k\) be the distinct positive divisors of the positive integer \(n\), including 1 and \(n\). Prove that \[(d_1 d_2 ... d_k)^2 = n^k.\]

The equation of the tangent plane to the real ellipsoid \[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\] at the point \((x_1, y_1, z_1)\) is \[\frac{xx_1}{a^2} + \frac{yy_1}{b^2} + \frac{zz_1}{c^2} = 1.\] Prove that the common tangent planes to the three ellipsoids \[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,\] \[\frac{x^2}{b^2} + \frac{y^2}{c^2} + \frac{z^2}{a^2} = 1,\] \[\frac{x^2}{c^2} + \frac{y^2}{a^2} + \frac{z^2}{b^2} = 1\] touch a sphere of radius \(\{(a^2 + b^2 + c^2)/3\}^{\frac{1}{2}}\), and that the points of contact of these planes with the ellipsoids lie on a sphere of radius \((a^4 + b^4 + c^4)^{\frac{1}{2}}(a^2 + b^2 + c^2)^{-\frac{1}{2}}\).

Let \(\cal S\) be an infinite set of pairs of points in the plane such that the points in question do not all lie on a circle or on a straight line. Show that the following two conditions on \(\cal S\) are equivalent: (i) there is a fixed point \(P\) and a constant \(k\) such that for all pairs \(\{X, X'\}\) in \(\cal S\), \(P\) lies on the line segment \(XX'\) and \(XP.PX' = k^2\); (ii) the four points of any two pairs in \(\cal S\) lie on a circle or line, in an ordering in which the pairs are interleaved. \(T\) is a transformation of the plane \(\Pi\), with an inverse \(T^{-1}\), such that both \(T\) and \(T^{-1}\) send all circles to circles. Let \(\mathcal{C}(P, k)\) be the set of all circles in \(\Pi\) containing a chord \(XX'\) such that \(P\) lies on the segment \(XX'\) and \(XP.PX' = k^2\). Show that \(T\) maps \(\mathcal{C}(P, k)\) to a set of circles of the form \(\mathcal{C}(\tilde{P}, \tilde{k})\). Show that \(T\) can be extended to a transformation of three-dimensional space containing \(\Pi\), that maps all spheres with centres in \(\Pi\) to spheres with centres in \(\Pi\).

Show that \(e^{-t^2/2} \geq \cos t\) for \(0 \leq t \leq \frac{1}{4}\pi\).

Let \(\omega = e^{\pi i/k}\), where \(k\) is an integer greater than 1. Let \(T_0 = 0\) and \[T_j = \omega + \omega^2 +...+ \omega^j.\] Show that \(T_{2k} = 0\), and sketch the polygon \(T_0 T_1...T_{2k}\) in the Argand diagram. Now let \(S_0 = 0\) and \(S_j = \omega + \omega^2/2 +...+ \omega^j/j\). Express \(S_j\) in terms of \(T_0, ..., T_j\) and show that each of the numbers \(S_0, ..., S_{2k}\) lies within or on the polygon \(T_0 T_1...T_{2k}\).

The function \(\log^+ (x)\) is defined by \[\log^+ (x) = \begin{cases} \log_e (x) & (x \geq 1) \\ 0 & (x < 1) \end{cases}\] Positive numbers \(\lambda_1 > \lambda_2 > ... > \lambda_n\) and \(\mu_1 > \mu_2 > ... > \mu_n\) satisfy \[\lambda_1 \lambda_2 ... \lambda_j \geq \mu_1 \mu_2 ... \mu_j \quad \text{for} \quad 1 \leq j \leq n.\] Show that \[g(x) = \sum_{j=1}^{n} \log^+ (\lambda_j x) \geq h(x) = \sum_{j=1}^{n} \log^+ (\mu_j x),\] for all \(x\). By considering \[\int_0^{\infty} \frac{g(x)}{x^{s+1}} dx \quad \text{and} \quad \int_0^{\infty} \frac{h(x)}{x^{s+1}} dx,\] show that \[\lambda_1^s + ... + \lambda_n^s \geq \mu_1^s + ... + \mu_n^s, \quad \text{for} \quad s > 0.\]

\(X\) is an integer-valued random variable, with distribution given by \[\text{Pr}[X = k] = \frac{c}{k \cdot 2^k}, \quad k \geq 1.\] Find the probability generating function of \(X\), and hence deduce the value of \(c\). A car insurance company observes that the number \(N\) of claims in any year is distributed as a Poisson random variable with mean \(\mu\), and that the sums of money paid out on the different claims are distributed, independently of \(N\) and of each other, in the same way as \(X\). By considering probability generating functions, or otherwise, find the mean and variance of the total sum \(S\) paid out per year. [Hint. If the probability generating function of \(S\), given that \(N = n\), is denoted by \(G_n(z)\), then the probability generating function of \(S\) is given by \(\sum_{n \geq 0} G_n(z) \cdot \text{Pr}[N = n]\).]