\hfill \thepage} %} \input{tcilatex} \begin{document} \section{Twenty-fourth International Olympiad, 1983} 1983/1. 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) $f(x)\rightarrow 0$ as $x\rightarrow \infty .$ 1983/2. 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 $\angle O_{1}AO_{2}=\angle M_{1}AM_{2}$. 1983/3. 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. 1983/4. 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. 1983/5. 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. 1983/6. Let $a,b$ and $c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a-b)+b^{2}c(b-c)+c^{2}a(c-a)\geq 0. \] Determine when equality occurs. \end{document}