IMO 1986

Let $d$ be any positive integer not equal to $2, 5$ or $13$ . Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$ . Define $A_s=A_{s-3}$ for all $s\ge4$ . Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$ . Prove that if $P_{1986}=P_0$ , then the triangle $A_1A_2A_3$ is equilateral.

To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$ , then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

Let $A,B$ be adjacent vertices of a regular $n$ -gon ( $n\ge5$ ) with center $O$ . A triangle $XYZ$ , which is congruent to and initially coincides with $OAB$ , moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$ .

Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x)\ne0$ for $0\le x<2$ , and $f(xf(y))f(y)=f(x+y)$ for all $x,y$ .

Given 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 any straight line $L$ parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on $L$ is not greater than $1$ ?