International Mathematical Olympiad 1983

Find all functions $f$ defined on the set of positive real numbers which take positive real values and satisfy the conditions:

(i) $f(xf(y)) = yf(x)$ for all positive $x, y$;

(ii) $f(x) \rightarrow 0$ as $x \rightarrow \infty$.

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$, respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$, and $M_2$ be the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2 = \angle M_1AM_2$.

Let $a, b$ and $c$ be positive integers, no two of which have a common divisor greater than 1. Show that $2abc - ab - bc - ca$ is the largest integer which cannot be expressed in the form $xbc + yca + zab$, where $x, y$ and $z$ are non-negative integers.

Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$ and $CA$ (including $A$, $B$ and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Justify your answer.

Is it possible to choose 1983 distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression? Justify your answer.

Let $a$, $b$ and $c$ be the lengths of the sides of a triangle. Prove that $$a^2b(a - b) + b^2c(b - c) + c^2a(c - a) \geq 0.$$

Determine when equality occurs.