On a chess board, which consists of 64 squares, a bishop is only allowed to move diagonally. In order to perform an '\(n\)-bounce' the Bouncing Bishop chooses a diagonal direction in which to travel and moves for \(n\) squares in such a way that on reaching the edge of the board he reverses the component of his motion perpendicular to that edge. For example, it is possible in one 6-bounce to travel from the intersection of the fourth row and fifth column to the intersection of the fourth row and third column or from the intersection of the fourth row and fourth column back to the intersection of the fourth row and fourth column. Show that, if \(n = 14k+t\) (or similarly \(14k-t\)), the effect of an \(n\)-bounce depends only on \(t\) and the initial direction of motion. Two squares are called \(n\)-equivalent if it is possible to travel from one to the other in a series of \(n\)-bounces. For each \(n\) find the number of \(n\)-equivalence classes of squares. (The \(n\)-equivalence class of a square is the set of squares which are \(n\)-equivalent to it.)
\(C\) is a closed, differentiable curve which is convex (i.e. any chord cuts it only twice). Points \(P\) and \(P'\) move round \(C\) in an anti-clockwise sense in such a way that the chord \(PP'\) has fixed length \(2a\); see Fig. 1. Show that the following properties are equivalent, in the sense that if \(C\) has any one of them it has all of them:
Let \(P_1 P_2 \ldots P_n\) be a regular polygon. Construct points \(Q_1\), \(Q_2\), \(\ldots\), \(Q_n\) such that \[\overrightarrow{P_1Q_1} = \overrightarrow{P_2P_3}, \overrightarrow{P_2Q_2} = \overrightarrow{P_3P_4}, \ldots, \overrightarrow{P_{n-1}Q_{n-1}} = \overrightarrow{P_nP_1}, \overrightarrow{P_nQ_n} = \overrightarrow{P_1P_2}.\] Prove that, if \(n \neq 7\) or if \(n = 5\), then \(Q_1 Q_2 \ldots Q_n\) is another regular polygon of smaller size. Deduce that, for these values of \(n\), it is impossible to embed a regular polygon in the integral plane lattice, i.e. with each vertex at a point whose Cartesian coordinates are integers. Prove further that the same result is also true for the equilateral triangle and regular hexagon, by showing that any such embedded polygon must have an area equal to a rational number, or otherwise.
Two circular non-overlapping discs lie in a given triangle \(ABC\). We wish to maximise the sum of their areas. Show first that, for a maximum, the boundary of each disc must touch the boundary of the other and (at least) two sides of the triangle; and hence that we may restrict consideration to cases such as that illustrated. [DIAGRAM MISSING] Prove that, in such a position, if \(x\), \(y\) denote the radii of the discs, then they satisfy \[4xy = (a-x\cot\alpha-y\cot\beta)^2,\] where \(\alpha\), \(\beta\) and \(a\) are as in the diagram. Deduce that, if \(y\) is plotted against \(x\), the relevant part of the graph is part of an ellipse, and that this ellipse touches the coordinate axes. Show further that the part of the ellipse in question is a subset of the shorter of the two arcs determined by the points of contact with the axes, by showing that the point with coordinates \((r, r)\) lies inside the ellipse, where \(r\) is the in-radius of the triangle \(ABC\). Hence show that the desired area \(\pi(x^2 + y^2)\), is maximised only when one or other of the discs is the incircle of the triangle \(ABC\), and the other touches it and just two of the sides of the triangle. Which pair of sides should we pick to obtain the maximum, if we are given that \(\hat{A} > \hat{B} > \hat{C}\)? Justify your answer.
Show that the complex mapping \(w = z+z^{-1}\), where \(z = x+iy\), \(w = u+iv\) are complex numbers, maps straight lines through \(z = 0\) into confocal hyperbolas in the \(w\)-plane with foci at \(w = \pm 2\); and maps each of the circles \(|z| = r\), \(|z| = r^{-1}\) into the same ellipse, also with foci \(\pm 2\). Prove that this ellipse cuts each such hyperbola orthogonally.
Two real differentiable functions \(u(x)\), \(v(x)\) are said to be linearly dependent in \(-1 \leq x \leq 1\) if there exist real constants \(\lambda\), \(\mu\), not both zero, such that \(\lambda u(x) + \mu v(x) = 0\) for all \(x\) in the range. Show that, if \(u(x)\), \(v(x)\) are linearly dependent, then each of the determinants \[D_1(x) = \begin{vmatrix} u(x) & v(x) \\ \frac{du}{dx} & \frac{dv}{dx} \end{vmatrix},\] \[D_2 = \begin{vmatrix} \int_{-1}^1 \{u(x)\}^2dx & \int_{-1}^1 u(x)v(x)dx \\ \int_{-1}^1 u(x)v(x)dx & \int_{-1}^1 \{v(x)\}^2dx \end{vmatrix}\] is zero. Prove that the converse `\(D_1(x) = 0\) for all \(x\) implies that \(u(x)\), \(v(x)\) are linearly dependent in \(-1 \leq x \leq 1\)' is false, by exhibiting an example of two functions \(u(x)\), \(v(x)\), differentiable at each point of the range, yet with one of them vanishing identically in the part \(-1 \leq x \leq 0\) of the range and the other vanishing identically in the part \(0 \leq x \leq 1\) of the range. Prove that the converse `\(D_2 = 0\) implies that \(u(x)\), \(v(x)\) are linearly dependent' is however true, by considering \[\int_{-1}^1 \{u(x)-\theta v(x)\}^2dx\] as a quadratic expression in \(\theta\).
\(S\) is a set of \(n\) points \(P_1\), \(P_2\), \(\ldots\), \(P_n\) equally spaced round the periphery of a circle (i.e. at the vertices of a regular polygon). \(A\) is a subset of them, \(a\) in number; \(B\) is another, \(b\) in number. \(\chi_A\), the 'characteristic function of \(A\)' is a function defined only on the points of \(S\), thus: \(\chi_A(P_i) = 1\) if \(P_i\) belongs to \(A\), \(\chi_A(P_i) = 0\) otherwise; \(\chi_B\) is defined similarly. \(B_k\) is the set obtained by rotating \(B\) through an angle \(2k\pi/n\) positively about the centre of the circle (so that \(B_k\) is also a subset of \(S\), and is congruent to \(B\)). By considering the double sum \[\sum_{k=1}^n \sum_{i=1}^n \chi_A(P_i)\chi_{B_k}(P_i),\] show that there exists a value of \(k\) for which \(A \cap B_k\) contains at least \(ab/n\) points.
In one game of a tennis match the probability that a player serving wins any particular point is \(\frac{3}{4}\). What is the probability that the player serving wins the game? [The game finishes as soon as one player has won at least four points and is at least two points ahead of his opponent.]
Let \(X\) and \(Y\) be two discrete random variables with correlation coefficient \(\rho(X, Y)\). Prove that \(|\rho(X, Y)| \leq 1\). Prove also that, if \(X\) and \(Y\) are independent, then \(\rho(X, Y) = 0\). Show that the converse of the latter result is true if \(X\) and \(Y\) take only two values; and show by giving an example that it is not true in general.
Two weights \(W_1\) and \(W_2\) are attached to the ends of a rope (of negligible weight) which is passed over a fixed rough horizontal cylinder of circular cross-section. By considering the forces on an element of rope in contact with the cylinder when the friction is limiting, find the manner in which the tension in the rope varies along its length, and hence show that static equilibrium is possible only if \[e^{-\mu\pi} \leq W_1/W_2 \leq e^{\mu\pi},\] where \(\mu\) is the static coefficient of friction between rope and cylinder. How is this result modified if the rope is coiled round the cylinder \(n\) times, the weights \(W_1\) and \(W_2\) being still suspended from its ends?