Problems

A point moves on the (fixed) set of points in the plane having integer coordinates \((m, n)\) with \(m \geq n\). The point starts at the origin \((0, 0)\) and at each step can move either one unit in the \(x\)-direction or one unit in the \(y\)-direction (provided this is possible); thus from \((m, n)\) it goes either to \((m + 1, n)\) or to \((m, n + 1)\) if \(m > n\), while it necessarily goes to \((m+1,n)\) if \(m = n\). Let \(h_p\) denote the number of distinct paths from \((0, 0)\) to \((p,p)\) for \(p \geq 0\) (so that \(h_0 = 1\)), and let \(k_p\) denote the number of distinct paths from \((0, 0)\) to \((p,p)\) for \(p > 0\) that do not include any of the points \((m, m)\) for \(0 \leq m \leq p - 1\). Show that \(h_p = \sum_{q=1}^{p} k_q h_{p-q}\) and that \(k_p = h_{p-1}\) for \(p \geq 1\). Writing \(H(x) = \sum_{p=0}^{\infty} h_p x^p\) and \(K(x) = \sum_{p=1}^{\infty} k_p x^p\), obtain an expression for \(H(x)\) as a function of \(x\), and hence show that \(h_p = \frac{1}{p+1}\binom{2p}{p}\).

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The points \(A,...,J\) are called nodes; two nodes are adjacent if they are joined by a straight line in the figure. Prove that each node belongs to just two collections of four mutually non-adjacent nodes. Show that there are five such collections, any two of which have just one node in common. Deduce that the permutations of the nodes which preserve adjacency form a group isomorphic to \(S_5\), the group of permutations of five objects.

Show that the equations in \(x_1, x_2, ..., x_n\) (with \(u, v\) constants): \[ux_1 x_2 + x_2 = v,\] \[ux_2 x_3 + x_3 = v,\] \[\vdots\] \[ux_{n-1} x_n + x_n = v,\] \[ux_n x_1 + x_1 = v,\] possess either one or two solutions with \(x_1 = x_2 = ... = x_n\), or else possess infinitely many solutions. Show that there are infinitely many solutions if and only if \(\sqrt{(1 + 4uv)}\) is a nonzero root of the equation \[(1+t)^n-(1-t)^n = 0,\] and hence if and only if \(uv = -\frac{1}{4} \sec^2 (\pi k/n)\) with \(1 \leq k \leq n - 1\).

Show that the composition of any two maps of the form \[z \to z_1 = \frac{az+b}{cz+d} \quad (a,b,c,d \text{ integers}; ad-bc=1)\] is of the same form and that the inverse of any map of this form is of the same form. Write down a formula for \(\operatorname{im} z_1\) involving \(\operatorname{im} z\) and show that any map of the form \(z \to z_1\) maps the upper half-plane \(H = \{z: \operatorname{im} z > 0\}\) in the complex plane onto itself. Show that any map of the form \(z \to z_1\) is a composition of maps of the form \(z \to z + n\) (\(n\) integer) and \(z \to -1/z\).

Let \(l\) be a fixed line in the plane. Let \(P\), \(Q\) be distinct points not on \(l\) lying on the same side of \(l\), and let \(\overline{P}\) be the reflexion of \(P\) in \(l\). Prove that there is a unique circle \(C\) passing through \(Q\) such that \(P\) and \(\overline{P}\) are inverse points with respect to \(C\). Define \(d(P, Q) = (\text{radius of }C)/P\overline{P}\). Prove that if \(P, Q\) are mapped to \(P', Q'\) by inversion in some circle with centre on \(l\), then \(d(P', Q') = d(P, Q)\). Deduce that \(d(Q, P) = d(P, Q)\). Let \(m\) be a line meeting \(l\). For which point \(X\) on \(m\) is \(d(P, X)\) minimum?

Let \(A\), \(B\), \(C\), \(D\) be fixed points in the plane, no three being collinear. Prove that the centres of the conics through \(A\), \(B\), \(C\), \(D\) all lie on a conic \(S\). Prove that \(S\) passes through the mid-point of \(AB\) and also through the intersection of \(AB\) and \(CD\). Show that \(S\) is a rectangular hyperbola if and only if \(A\), \(B\), \(C\), \(D\) lie on a circle.

Let \(S\) be the surface of a sphere of unit radius. The intersection of \(S\) with a plane through its centre is called a great circle. Let \(\Delta\) be a curvilinear triangle on \(S\) whose edges are arcs of great circles \(C_1, C_2, C_3\). By considering the areas of all the regions into which \(C_1, C_2, C_3\) divide \(S\), or otherwise, show that the sum of the angles of \(\Delta\) is \(\pi +\) area of \(\Delta\). A convex polyhedron with triangular faces has \(v\) vertices, \(e\) edges and \(f\) faces. Show that \(e = \frac{3f}{2}\) and \(v-e+f = 2\).

The function \(f\) satisfies the equation \[f(x) = \frac{1}{4}\left(f\left(\frac{x}{2}\right)+f\left(\frac{x+\pi}{2}\right)\right)\] for \(0 < x < \pi\). Show that if there is a constant \(M\) such that \(|f(x)| < M\) for \(0 < x < \pi\), then \(f(x) = 0\) whenever \(0 < x < \pi\). Given that \[\sum_{n=1}^{\infty} \frac{1}{(x-n\pi)^2} < 1\] for \(|x| \leq \frac{\pi}{2}\) and \(\textrm{cosec}^2 x - \frac{1}{x^2} < 1\) for \(0 < x < \frac{\pi}{2}\), prove that \[\textrm{cosec}^2 x = \sum_{n=-\infty}^{\infty} \frac{1}{(x-n\pi)^2}\] whenever \(x\) is not a multiple of \(\pi\).

The function \(f\) satisfies \(f(-y) = -f(y)\) and is defined as follows for \(y \geq 0\). \[f(y) = y \quad \text{if } 0 \leq y \leq 1,\] \[f(y) = 2-y \quad \text{if } 1 \leq y \leq 2,\] \[f(y) = 0 \quad \text{if } y \geq 2.\] Solve the differential equation \(y'' + f(y) = 0\) with initial conditions \(y(0) = 0\), \(y'(0) = c\). Sketch the solutions corresponding to initial conditions \(y(0) = 0\), \(y'(0) = c\) for \(c = 1\), \(c = \frac{4}{3}\) and \(c = \frac{8}{3}\).

A standard pack of 52 cards is thoroughly shuffled, and then dealt into four piles as follows. Cards are dealt into the first pile up to and including the first ace, then into the second pile up to and including the second ace, then into the third pile up to and including the third ace, then into the fourth pile up to and including the fourth ace, and then any remaining cards go into the first pile again. A second similar pack is thoroughly shuffled, and a single card drawn from it at random. Find the probability distribution of the size of the pile that contains the matching card from the first pack.