The letters \(n\) and \(k\) denote positive integers.
Prove that the geometric mean of \(n\) positive numbers cannot exceed their arithmetic mean. Deduce that if \(x, y, z\) are positive numbers such that \(x + y + z = 1\), and \(a, b, c\) are positive integers, then $$x^a y^b z^c \leq a^a b^b c^c / (a + b + c)^{a+b+c}.$$
The equality $$\frac{ax^2 + bx + c}{(x + \alpha)(x + \beta)(x + \gamma)} = \frac{A}{(x + \alpha)} + \frac{B}{(x + \beta)} + \frac{C}{(x + \gamma)},$$ in which \(a, b, c, \alpha, \beta, \gamma\) are real numbers, holds for all real \(x\) (other than \(-\alpha, -\beta, -\gamma\)). Find a necessary and sufficient condition, in terms of \(a, b, c\), for \(A + B + C\) to vanish. Evaluate $$\sum_{n=1}^{N} \frac{2n + 5}{n(n + 1)(n + 3)} \quad\text{and}\quad \sum_{n=1}^{\infty} \frac{2n + 5}{n(n + 1)(n + 3)}.$$
\(Z\) denotes the set of all integers, positive, negative, and zero. An equivalence relation \(R\) on \(Z\) is said to be a congruence relation if there exists an integer \(d \geq 0\) such that \(xRy\) if and only if \(x - y\) is an integral multiple of \(d\). Show that an equivalence relation \(S\) on \(Z\) which is such that \((x - y)S(z - w)\) whenever \(xSz\) and \(ySw\) is a congruence relation. Find an equivalence relation \(T\) on \(Z\) which is not a congruence relation but is such that \((xy)T(zw)\) whenever \(xTz\) and \(yTw\).
An element \(x\) of a finite multiplicative group \(G\), with identity \(e\), is said to have finite order if \(x^n = e\) for some positive integer \(n\). The order of \(x\) is the least such integer. Show that every element of \(G\) has finite order. If \(x\) and \(y\) are elements of \(G\), show that \(x\) and \(y^{-1}xy\) have the same order. If \(G\) has only one element \(z\) of order 2, show that \(z\) commutes with every element \(w\) of \(G\), that is, that \(zw = wz\).
Prove that if \(z\) and \(w\) are complex numbers then $$\arg(zw) = \arg(z) + \arg(w)$$ to within a multiple of \(2\pi\). Deduce that if \(z_1, \ldots, z_n\) are \(n\) complex numbers such that $$0 < \arg(z_r) < \frac{2\pi}{n-1} \quad\text{for } r = 1, 2, \ldots, n$$ then, provided that \(n > 2\), we have \(z_1 + \cdots + z_n \neq z_1 z_2 \cdots z_n\).
If \(p\) is a prime number and \(\omega \neq 1\) is a complex root of the equation \(z^p = 1\), how are the roots of \(1 + z + \cdots + z^{p-1} = 0\) related to \(\omega\)? Justify your answer. Let \(f\) be a polynomial with integral coefficients and degree greater than 1. By considering \(f(z)f(z^2)f(z^3)\cdots f(z^{p-1})\) show that if \(f(1) = 1\) then \(f(\omega) \neq 0\).
Show that the group of all rotations of a cube onto itself is isomorphic to the group of all permutations of four letters \(a, b, c, d\), by considering the effect of rotations on the four main diagonals \(AA', BB', CC', DD'\) of any rotation (see the Figure). By considering the effects of rotations of the cube on the three axes \(EE', FF', GG'\) show also that with each permutation \(\pi\) of \(a, b, c, d\) we can associate a permutation \(\pi'\) of the three letters \(e, f, g\) in such a way that \((\pi_1 \pi_2)' = \pi_1' \pi_2'\) and that every possible permutation of \(e, f, g\) appears among the permutations \(\pi'\).
\(P\) and \(Q\) are points of the plane outside the circumcircle of the regular polygon \(A_0 A_1 A_2 \ldots A_{n-1}\) whose centre is the point \(O\). The line-segment \(OP\) contains a vertex of the polygon, while the segment \(OQ\) perpendicularly bisects an edge. By representing this situation in the complex plane, or otherwise, show that the geometric mean of all the lengths \(QA_r\) exceeds the length \(QO\), while the length \(PO\) exceeds the geometric mean of all the lengths \(PA_r\).
\(C\) is a circle whose centre is a point \(P\) on a rectangular hyperbola \(R\), and which passes through the centre \(O\) of \(R\). \(T\) is the diameter of \(C\) which is tangent to \(R\) at \(P\), and \(N\) is the diameter which is normal to \(R\) at \(P\). Show that the endpoints of \(T\) lie on the asymptotes of \(R\), while those of \(N\) lie on the axes of symmetry.