A pendulum consists of a bob of mass \(M\) suspended by a light string of length \(l\) from a point that is forced to move along a horizontal straight line with displacement \(x(t)\). The air exerts a resistive force on the bob equal to \(kM\) times its speed. Find the exact equation of motion, and show that for small angular deviations \(\theta\) from the vertical it is approximately $$l\ddot{\theta} + kl\dot{\theta} + g\theta = -(\ddot{x} + k\dot{x}).$$ Show that, if \(x = a\cos\omega t\) where \(\omega^2 = g/l\) and \(a\) is small (so that the approximate equation of motion may be used), an (undamped) periodic motion is possible. Determine \(\theta(t)\) for this motion, and calculate the energy dissipated in one complete swing of the pendulum.
Five equal uniform bars, each of mass \(M\), are freely jointed together to form a plane pentagon \(ABCDE\). They are suspended from \(A\), and are constrained by equal light strings \(AC\) and \(AD\) so as to form a regular pentagon. Show without direct calculation that the tension in each string is the same as it would be if the bars were replaced by light rods and a mass \(M\) attached at each vertex. Hence show that this tension has magnitude \(2Mg\cos\frac{1}{5}\pi\).
If a (commutative) ring has multiplicative identity 1, the element \(x\) is said to have order \(n\) if \(n\) is the least positive integer for which \(x^n = 1\). Show, by considering the elements \(-1\) and \(1 + u + u^4\), that, if a ring has an element \(u\) of order 5, then it has either an element of order 2, or one of order 3. [Note: It is possible to have \(1 + 1 = 0\) in a ring.]
The sequence \(a_0, a_1, \ldots, a_{n-1}\) is such that, for each \(i\) \((0 \leq i \leq n-1)\), \(a_i\) is the number of \(i\)'s in the sequence. (Thus for \(n = 4\) we might have \(a_0, a_1, a_2, a_3 = 1, 2, 1, 0\).) If \(n \geq 7\), show that the sequence can only be $$n-4, 2, 1, 0, 0, \ldots, 0, 1, 0, 0, 0.$$ [Hint: Show that the sum of all the terms is \(n\), and that there are \(n - a_0 - 1\) non-zero terms other than \(a_0\), which sum to \(n - a_0\).]
The famous Four Colour Theorem (still unproved) asserts that the regions of any geographical map in the plane may be coloured using only four colours in such a way that regions which touch along an edge are distinctly coloured. In the case when there is a path composed of edges which includes every vertex just once, show that it is possible to colour the map in such a way that two colours are used for the portion enclosed by the path, and two for the remainder. [You may suppose that the regions of the map are straight-edged polygons, whose edges and vertices are called the edges and vertices of the map.]
If \(A\), \(B\), \(C\) are numbers such that \(A t^2 + 2Bt + C \geq 0\) for all real \(t\), show that \(B^2 \leq AC\). By considering \((f(x) + g(x))^2\), show that $$\left(\int_a^b f(x)g(x)dx\right)^2 \leq \int_a^b (f(x))^2 dx \int_a^b (g(x))^2 dx$$ for any continuous functions defined on the interval \([a, b]\). Obtain the inequality $$\int_0^{\pi/2} \sin^4 x \, dx \leq \frac{1}{8}\sqrt{\pi}.$$
The one-player game of Topswaps is played as follows. The player holds a pack of \(n\) cards, numbered from 1 to \(n\) in a random order. If the top card is numbered \(k\), he calls \(k\), reverses the order of the top \(k\) cards, and continues. Show that the pack eventually reaches a constant state in which the top card is numbered 1. [Hint: if \(k > 1\), and, from some point onwards, no card numbered higher than \(k\) is called, then \(k\) is called at most once thereafter.]
Let \(n\) be an odd number such that some power of 2 leaves remainder 1 on division by \(n\). Show, by considering the sequence of remainders of \(1, 2, 2^2, \ldots\) on division by \(n\), that there is a number \(m < n\) such that \(2^k - 1\) is divisible by \(n\) if and only if \(k\) is divisible by \(m\). If \(2^n - 1\) is divisible by \(n\), show that \(2^m - 1\) is divisible by \(m\). Deduce that for no number \(n\) greater than 1 is \(2^n - 1\) divisible by \(n\).
\(ABC\) is a triangle, whose angles are \(3\alpha, 3\beta, 3\gamma\). Points \(P, Q, R\) interior to the triangle are such that \begin{align} \angle PBC &= \beta, \quad \angle PCB = \gamma,\\ \angle PBC &= \beta, \quad \angle CRQ = \frac{1}{3}\pi + \beta,\\ \angle PCQ &= \gamma, \quad \angle BPR = \frac{1}{3}\pi + \gamma. \end{align} The points \(H\), on \(AC\), and \(K\), on \(AB\), are such that \(\angle QHC = \frac{1}{3}\pi + \beta\), \(\angle RKB = \frac{1}{3}\pi + \gamma\). Prove (i) that the triangle \(PQR\) is equilateral, (ii) that \(A, K, R, Q, H\) lie on a circle, and (iii) that \(AR, AQ\) trisect the angle \(A\).
The number of hours of sleep of a group of patients was recorded. On a subsequent night the patients were each given a sleeping pill and the number of hours of sleep was again recorded. The results were as follows: