Problems

1. Show that the product of three consecutive positive integers is divisible by 60 if the middle one is a square, and by 240 if the middle one is an odd square.
2. Show that if \(n\) is a non-negative integer, \(4^{3n} + 5^{2n+2}\) cannot be a prime.

Prove that the geometric mean of \(n\) positive numbers cannot exceed their arithmetic mean. Deduce that if \(x\), \(y\) and \(z\) are positive numbers such that $$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1,$$ then $$(x-1)(y-1)(z-1) \geq 8.$$

Evaluate

1. \(\sum_{r=1}^{n} \frac{r-1}{r(r+1)(r+2)}\) \quad \((n \geq 3)\);
2. \(1^2 + 2^2c_1 + 3^2c_2 + \ldots + (n+1)^2c_n\), where \(c_r = \frac{n!}{r!(n-r)!}\).

A 3-inch square tile is decorated by dividing one face into 9 equal squares, and painting the resulting 1-inch squares red, green, or yellow, in such a way that no two squares with a common edge are painted the same colour. How many different tiles are possible?

For what values of \(a\), \(b\) and \(c\) are the following equations consistent? \begin{align} x + y + z &= 1, \\ ax + by + cz &= 0, \\ a^2x + b^2y + c^2z &= 0. \end{align} Solve them completely when they are consistent.

1. State, giving adequate reasons, whether the following sets, with the given operations, are groups.
   1. The integers 1, 2, 3 and 4, under ordinary multiplication, reduced modulo 5.
   2. Rationals of the form \(p/q\), where \(p\) and \(q\) are integers with \(q > 0\) and \(q\) prime to 3, under addition.
   3. Real \(2 \times 2\) matrices, under matrix multiplication.
2. Does the set of 3-dimensional real vectors form a ring under vector addition and vector multiplication? Justify your answer.

A relation \(R\) between elements \(a\), \(b\), \(\ldots\) of a group \(G\) is defined by the rule ``\(aRb\) if and only if there exists \(g \in G\) such that \(b = gag^{-1}\)''. Show that \(R\) is an equivalence relation. For a fixed element \(a \in G\), a second relation \(S\) between elements \(g\), \(h\), \(\ldots\) of \(G\) is defined by the rule ``\(gSh\) if and only if \(gh^{-1}a = agh^{-1}\)''. Show that \(S\) is also an equivalence relation, and that there is a (1--1) correspondence between the set of equivalence classes under \(R\) and the set of elements in the equivalence class of \(a\) under \(R\).

Three distinct complex numbers \(z_1\), \(z_2\), \(z_3\) are represented in the complex plane by points \(A_1\), \(A_2\), \(A_3\). Prove that a necessary and sufficient condition for the triangle \(A_1A_2A_3\) to be equilateral is $$z_1^2 + z_2^2 + z_3^2 = z_2z_3 + z_3z_1 + z_1z_2.$$

The tangents at two points \(A\), \(A'\) of a circle \(S\) meet in \(T\). The mid-points of \(TA\), \(TA'\) are \(L\), \(L'\). \(P\) is any point of \(S\) and the lines \(PA\), \(PA'\) meet the line \(LL'\) in \(Q\), \(Q'\). Prove that \(QT^2 = QA \cdot QP\) and that the points \(T\), \(P\), \(Q\), \(Q'\) are concyclic.

The rectangular hyperbola \(xy = k^2\) is met by a circle passing through its centre \(O\) in four points \(A_1\), \(A_2\), \(B_1\), \(B_2\). The lengths of the perpendiculars from \(O\) to \(A_1A_2\) and \(B_1B_2\) are \(a\) and \(b\). Prove that $$ab = k^2.$$