IMO 1983

Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$ ; and $f(x)\to0$ as $x\to\infty$ .

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 $M_2$ 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 $x,y,z$ 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.

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?

Let $a$ , $b$ and $c$ be the lengths of the sides of a triangle. Prove that $$a^{2}b(a - b) + b^{2}c(b - c) + c^{2}a(c - a)\ge 0.$$ Determine when equality occurs.