International Mathematical Olympiad · 1986

Problems

Problem 1Let $d$ be any positive integer not equal to $2$ , $5$ , or $13$ . Show that one can find distinct elements $a$ and $b$ in the set $\{2, 5, 13, d\}$ such that $ab - 1$ is not a perfect square.Solutions: 2
Problem 2Given a point $P_0$ in the plane of triangle $A_1A_2A_3$ , define $A_s=A_{s-3}$ for all $s\ge4$ . Construct points $P_1,P_2,\ldots$ so that $P_{k+1}$ is the image of $P_k$ under a clockwise rotation through $120^\circ$ about $A_{k+1}$ . Prove that if $P_{1986}=P_0$ , then triangle $A_1A_2A_3$ is equilateral.Solutions: 1
Problem 3To each vertex of a regular pentagon an integer is assigned, with positive total sum. If three consecutive vertices carry $x,y,z$ and $y<0$ , replace them by $x+y,-y,z+y$ . This operation is repeated whenever some number is negative. Must the procedure always end after finitely many steps?Solutions: 2
Problem 4Let $A,B$ be adjacent vertices of a regular $n$ -gon ( $n\ge5$ ) with center $O$ . A triangle $XYZ$ , congruent to and initially coinciding with $OAB$ , moves so that $Y$ and $Z$ each trace the whole boundary of the polygon, while $X$ remains inside the polygon. Find the locus of $X$ .Solutions: 1
Problem 5Find all functions $f$ mapping the non-negative reals onto the non-negative reals such that $f(xf(y))f(y)=f(x+y)$ for all non-negative reals $x,y$ , with $f(2)=0$ and $f(x)\ne0$ for every $0\le x<2$ .Solutions: 2
Problem 6Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for every line $L$ parallel to one of the coordinate axes, the absolute difference between the numbers of white and red points on $L$ is at most $1$ ?Solutions: 1