Necklaces consist of \(n + 3\) beads threaded on a loop of string, without a clasp and with a negligible knot, so that the beads may move round freely. Prove that the number of distinguishable necklaces that can be made from \(n\) blue beads, 2 red ones and 1 yellow one is $$\frac{1}{2}(n + 2)^2$$ if \(n\) is even. What is the corresponding number if \(n\) is odd?
(i) Prove that $$\frac{1}{4} - \frac{1}{n+1} < \sum_{r=4}^n \frac{1}{r^2} < \frac{1}{24} - \frac{2n+1}{2n(n+1)} \quad (n \geq 4).$$ (ii) Find \(\sum_{r=1}^n \frac{1}{rx^r}\).
Solve the linear recurrence relation $$u_n = (n-1)(u_{n-1} + u_{n-2}),$$ given that \(u_1 = 0\) and \(u_2 = 1\), by writing \(u_n = v_n \cdot n!\), or otherwise. Show that as \(n \to \infty\), \(u_n/n! \to 1/e\).
Prove that $$\begin{vmatrix} (a-x)^2 & (a-y)^2 & (a-z)^2 & (a-w)^2 \\ (b-x)^2 & (b-y)^2 & (b-z)^2 & (b-w)^2 \\ (c-x)^2 & (c-y)^2 & (c-z)^2 & (c-w)^2 \\ (d-x)^2 & (d-y)^2 & (d-z)^2 & (d-w)^2 \end{vmatrix} = 0.$$
\(a\), \(b\), \(c\) are three positive numbers. Prove the inequality $$abc \geq (b + c - a)(c + a - b)(a + b - c).$$ Is the inequality $$abc \geq \kappa(b + c - a)(c + a - b)(a + b - c)$$ true for any constant \(\kappa\) greater than 1 (and all \(a\), \(b\), \(c\))?
Solution: Unless \(a,b,c\) are the sides of a triangle, the LHS is positive and the RHS is negative. Therefore wlog, \(a = x+y, b = y+z, c = z+x\) (Ravi substitution, consider the incircle). Therefore it is sufficient to prove \begin{align*} && (x+y)(y+z)(z+x) &> 2z\cdot 2x\cdot 2y = 8xyz \end{align*} but \(x+y \geq 2\sqrt{xy}, y+z \geq 2\sqrt{yz}, z+x \geq 2\sqrt{zx}\) and so taking their product we obtain the required solution. No - plugging in \(a=b=c=1\) we see that \(1 \geq \kappa\).
Four complex numbers are denoted by \(z_1\), \(z_2\), \(z_3\), \(z_4\). Show that their representative points in the complex plane are concyclic if and only if the cross-ratio $$\frac{(z_1 - z_3)(z_3 - z_4)}{(z_1 - z_4)(z_3 - z_2)}$$ is real. Use this result to show that if these points are concyclic so are the points \(1/z_2\), \(1/z_3\), \(1/z_4\).
\(AB\), \(AC\) are two equal line segments, meeting at an acute angle. \(X\) is a point such that \(AB\), \(AC\) subtend equal angles at \(X\). Find the locus of \(X\). (Angles at \(X\) are to be reckoned positive in whichever sense they are measured.)
Find the \emph{width} of a regular tetrahedron of side \(a\), where \emph{width} is defined as the least distance between a pair of distinct parallel planes, each of which 'touches' the tetrahedron, either at a vertex, along an edge, or on a face. Hence or otherwise show that it is impossible to put a regular tetrahedron of side greater than \(\sqrt{2}\) in a cube of side 1. Show, with a sketch, how a regular tetrahedron of side exactly \(\sqrt{2}\) may be put in such a cube (the boundary of the cube may be used).
Prove that the locus, if it exists, of the meets of perpendicular real tangents to the hyperbola \(x^2/a^2 - y^2/b^2 = 1\) is a circle. Show that it cannot cut the hyperbola again.
\(p(\phi)\) is the positive length of the projection of a fixed line-segment of length \(l\) on an axis at a variable direction \(\phi\). Prove that $$\int_{\phi=0}^{\phi=2\pi} p(\phi) d\phi = 4l.$$ Hence or otherwise prove that if a triangle \(ABC\) lies entirely within a triangle \(XYZ\), then $$\text{perimeter } \triangle ABC < \text{perimeter } \triangle XYZ.$$