International Mathematical Olympiad · 1983

Problems

Problem 1Find all functions $f$ defined on the set of positive real numbers which take positive real values and satisfy the two conditions: (i) $f(xf(y)) = yf(x)$ for all positive real numbers $x, y$ ; (ii) $f(x) \to 0$ as $x \to \infty$ .Solutions: 1
Problem 2Let $A$ be one of the two distinct intersection points of unequal coplanar circles $C_1,C_2$ with centers $O_1,O_2$ . One common tangent touches them at $P_1,P_2$ , and the other at $Q_1,Q_2$ . Let $M_1,M_2$ be the midpoints of $P_1Q_1,P_2Q_2$ . Prove that $\angle O_1AO_2=\angle M_1AM_2$ .Solutions: 1
Problem 3Let $a,b,c$ be positive integers, no two having a common divisor greater than $1$ . Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed as $xbc+yca+zab$ , where $x,y,z$ are non-negative integers.Solutions: 1
Problem 4Let $ABC$ be an equilateral triangle and let $\mathcal E$ be the union of its three sides. Determine whether every partition of $\mathcal E$ into two disjoint subsets has one subset containing the vertices of a right-angled triangle. Justify your answer.Solutions: 1
Problem 5Is 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.Solutions: 1
Problem 6Let $a,b,c$ be the side lengths of a triangle. Prove that $a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge0$ . Determine when equality occurs.Solutions: 1