Let \(ABC\) be a triangle, \(P\), \(Q\), \(R\) the mid-points of \(BC\), \(CA\), \(AB\) respectively, and \(X_1\), \(X_2\), \(Y_1\), \(Y_2\), \(Z_1\), \(Z_2\) the mid-points of \(BP\), \(PC\), \(CQ\), \(QA\), \(AR\), \(RB\) respectively. If each of these nine mid-points is joined to the opposite vertex of the triangle, prove that in the resulting figure there are seven points inside the triangle through which three lines pass. Deduce that there exists a conic touching the lines \(AX_1\), \(AX_2\), \(BY_1\), \(BY_2\), \(CZ_1\), \(CZ_2\). (Standard theorems used need not be proved, but must be carefully stated.)
If the lengths of the sides of a quadrilateral are given, show that the quadrilateral has maximum area when it is cyclic.
A set of points \(S\) in the plane is called \emph{convex} if, for every pair of points \(P\), \(Q\) in \(S\), the line segment \(PQ\) lies in \(S\). Prove that the set of points whose coordinates \((x, y)\) satisfy $$y^2 \leq x \leq 1 - y^2$$ is convex. Give an example of a set which is not convex.
If a fair coin (i.e. one without bias) is tossed \(n\) times, show that the probability that \(r\) heads and \((n-r)\) tails occur is $$\frac{n!}{r!(n-r)!}2^{-n}.$$ An experimenter decides to continue tossing a fair coin until \(k\) heads have occurred. Find the probability \(p_n\) that he will have to perform exactly \(n\) tosses, and show that \(p_n\) is the coefficient of \(z^n\) in the power series expansion of $$z^k(2-z)^{-k}.$$ Deduce that $$\sum_{n=k}^{\infty} p_n = 1$$ and interpret this result.
\(a_0\), \(a_1\), \(\ldots\), \(a_{n-1}\) are complex numbers, and \(A_0\), \(A_1\), \(\ldots\), \(A_{n-1}\) are defined by $$A_s = \frac{1}{\sqrt{n}} \sum_{r=0}^{n-1} a_r \omega^{rs},$$ where \(\omega = e^{2\pi i/n}\). Prove that $$a_r = \frac{1}{\sqrt{n}} \sum_{s=0}^{n-1} A_s \omega^{-rs}$$ and that $$\sum_{s=0}^{n-1} |A_s|^2 = \sum_{r=0}^{n-1} |a_r|^2.$$
By considering the sum of the roots of the equation \(z^5 = 1\), find an equation with integer coefficients which is satisfied by \(\cos \frac{2\pi}{5}\), and hence obtain an expression for \(\cos \frac{2\pi}{5}\). Prove the theorem (known to Euclid) that if a pentagon, a hexagon, and a decagon, regular and with sides \(a_5\), \(a_6\), \(a_{10}\) are inscribed in the same circle, then $$a_5^2 = a_6^2 + a_{10}^2.$$
Criticize the following arguments: (i) \(\int \frac{d\theta}{5+4\cos\theta} = \int \frac{\sec^2 \frac{1}{2}\theta d\theta}{9+\tan^2 \frac{1}{2}\theta} = \frac{2}{3}\tan^{-1}(\frac{1}{3}\tan \frac{1}{2}\theta)\), \(\therefore \int_0^{2\pi} \frac{d\theta}{5+4\cos\theta} = \frac{2}{3}(\tan^{-1} 0 - \tan^{-1} 0) = 0\). (ii) The differential equation \(y'' + 2y'y = 0\) is satisfied by the functions \(y = 1/x\) and \(y = \cot x\); its general solution is therefore \(A \cot x + B/x\). Another solution is \(y = \tanh x\), therefore \(\tanh x\) is equal to a linear combination of \(\cot x\) and \(1/x\). Solve the differential equation completely.
A moving particle of mass \(M\) hits another particle of mass \(m\) which is at rest. The first particle goes on at an angle of \(30^{\circ}\) to its original track, but the subsequent path of the second particle is not observed. Supposing the collision is elastic, i.e. no kinetic energy is lost, prove (using vectors or otherwise) that \(M\) cannot have been more than \(2m\). It is later found that the particles were of equal mass, but that the collision may have been inelastic, some energy being taken up in producing internal motions in the particles. Find the greatest amount of energy that can have been absorbed in this way, as a fraction of the original kinetic energy of the first particle.
An airgun fires a shot of mass \(m\) vertically upwards, with velocity \(u\). In passing through the air its motion is resisted by a force equal to \(mg/u^2\), multiplied by the square of its velocity. Show that it will reach a height \(\frac{1}{2}(u^2/g)\log 2\), and find the velocity with which it hits the ground again.
A uniform pole of length \(2a\), standing vertically on rough ground, is slightly disturbed and begins to fall over. If it has not slipped by the time it makes an angle \(\theta\) with the vertical, show that $$\frac{d\theta}{dt} = \left(\frac{3g}{a}\right)^{1/2} \sin \frac{\theta}{2}.$$ Find the horizontal and vertical components of the force exerted by the ground on the pole, as a function of \(\cos\theta\), and prove that the pole will certainly have slipped before it can reach a certain angle \(\alpha\), however great the coefficient of friction may be.