IMO 1983

Problem 1. 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) $f(x)\to 0$ as $x\to \infty$.

Problem 2. 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 $P_1{Q}_1$,and $M_2$ be the midpoint of $P_2{Q}_2$. Prove that $\angle O_{1}A{O}_2=\angle M_{1}A{M}_2$.

Problem 3. 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.

Problem 4. 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.

Problem 5. 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.

Problem 6. 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.