Let \(x = x(t)\), \(y = y(t)\) be parametric equations for a simple closed curve \(C\) in the \(x, y\) plane, described counter-clockwise as \(t\) increases from \(t_0\) to \(t_1\). Show that the area \(A\) enclosed by \(C\) is given by \begin{equation*} A = -\int_{t_0}^{t_1} y(t)\frac{dx(t)}{dt} dt. \end{equation*} Hence show that \(\displaystyle A = -\frac{1}{2}\int_{t_0}^{t_1}\left[y(t)\frac{dx(t)}{dt} - x(t)\frac{dy(t)}{dt}\right] dt\). Use this result to find the area enclosed by the hypocycloid \(x^{2/3} + y^{2/3} = a^{2/3}\).
Find all positive integers that are equal to the sum of the squares of their digits.
State an inequality between the arithmetic mean of \(k\) positive numbers and their geometric mean. The numbers \(a_1, a_2, \ldots, a_n\) are positive. Assume that \(1 \leq k \leq n\) and let \(S_k\) be the sum of the \(k\)th powers of the numbers, and let \(P_k\) be the sum of all products of \(k\) distinct numbers from \(a_1, a_2, \ldots, a_n\). Prove that \begin{equation*} (n-1)! S_k \geq k!(n-k)! P_k. \end{equation*}
A magic square of order \(n \geq 3\) is an arrangement of the numbers 1 to \(n^2\) in a square so that the sum of the numbers in every row, in every column and in each long diagonal is the same. Prove that in a magic square of order \(n\), this common number is equal to \(\frac{1}{2}n(n^2+1)\). Show that in a magic square of order 3, 5 is in the centre, and 1 is not in a corner. Prove also that there are precisely two magic squares of order three in which 1 is in the middle of the top row.
A sequence \(a_0, a_1, a_2, \ldots\) is defined by the following recurrence relation: \begin{equation*} a_n = a_0a_{n-1} + a_1a_{n-2} + \ldots + a_{n-1}a_0, \quad a_0 = 1 \end{equation*} Setting \(f(x) = \sum_{n=0}^{\infty}a_nx^n\), show that \(f(x)\) satisfies the equation \begin{equation*} xf(x)^2 - f(x) + 1 = 0. \end{equation*} Deduce that \(a_n = \frac{1}{n+1}\binom{2n}{n}\). [The convergence of series may be assumed.]
For \(r = 1, 2, \ldots, n\) show that \(\binom{n}{r} < \frac{n^r}{2^{r-1}}\). If \(R_n = (1 + 1/n)^n\) show that, provided \(n \geq 2\), \(2 < R_n < 3\). Show also that \(R_{n+1} > R_n\).
Polynomials \(H_n(x)\) are defined by \begin{equation*} H_n(x) = (-1)^n e^{\frac{1}{2}x^2}\frac{d^n}{dx^n}(e^{-\frac{1}{2}x^2}). \end{equation*} Show that \(\sum_{n=0}^{\infty}\frac{1}{n!}H_n(x)y^n = \exp(xy - \frac{1}{2}y^2)\). Hence or otherwise show that \(\frac{dH_n}{dx}(x) = nH_{n-1}(x)\). [Taylor's theorem may be assumed. Questions of convergence need not be considered.]
The points \(A, B, C, D\) are vertices of a tetrahedron, with the origin at an internal point \(O\). The position vectors of \(A, B, C, D\) are then \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\), \(\mathbf{d}\). Show that the equation of the plane \(BCD\) may be written as \begin{equation*} \mathbf{r} = \beta\mathbf{b} + \gamma\mathbf{c} + \delta\mathbf{d}, \quad \beta + \gamma + \delta = 1. \end{equation*} Prove that there exist positive numbers \(p, q, r, s\) such that \begin{equation*} p\mathbf{a} + q\mathbf{b} + r\mathbf{c} + s\mathbf{d} = \mathbf{0}. \end{equation*} If the line \(AO\) intersects the plane \(BCD\) at \(E\), find the position vector of \(E\) and determine \(\frac{AO}{AE}\) in terms of \(p, q, r, s\). The lines \(AO, BO, CO, DO\) meet the opposite faces of the tetrahedron at \(E, F, G, H\) respectively. Show that \begin{equation*} \frac{AO}{AE} + \frac{BO}{BF} + \frac{CO}{CG} + \frac{DO}{DH} = 3. \end{equation*}
Bar magnets are placed randomly end to end in a straight line. If adjacent magnets have ends of different polarities facing each other, they join together to form a single unit. If they have ends of the same polarity facing each other, they stand apart. Find the expectation and variance of the number of separate units in terms of the total number \(N\) of magnets.