IMO 1986 — Official problems

The set $S=\{2,5,13\}$ has the property that for every $a,b\in S$, $a\ne b$, the number $ab-1$ is a perfect square. Show that for every positive integer $d$ not in $S$, the set $S\cup\{d\}$ does not have the above property.

Let $A,B,C$ be fixed points in the plane. A man starts from a certain point $P_0$ and walks directly to $A$. At $A$ he turns his direction by $60^\circ$ to the left and walks to $P_1$ such that $P_0A=AP_1$. After he performs the same action 1986 times successively around the points $A,B,C,A,B,C,\ldots$, he returns to the starting point. Prove that $ABC$ is an equilateral triangle, and that the vertices $A,B,C$ are arranged counterclockwise.

To each vertex $P_i$ ($i=1,\ldots,5$) of a pentagon an integer $x_i$ is assigned, the sum $s=\sum x_i$ being positive. The following operation is allowed, provided at least one of the $x_i$'s is negative: Choose a negative $x_i$, replace it by $-x_i$, and add the former value of $x_i$ to the integers assigned to the two neighboring vertices of $P_i$ (the remaining two integers are left unchanged).
This operation is to be performed repeatedly until all negative integers disappear. Decide whether this procedure must eventually terminate.

Let $A,B$ be adjacent vertices of a regular $n$-gon in the plane and let $O$ be its center. Now let the triangle $ABO$ glide around the polygon in such a way that the points $A$ and $B$ move along the whole circumference of the polygon. Describe the figure traced by the vertex $O$.

Find, with proof, all functions $f$ defined on the nonnegative real numbers and taking nonnegative real values such that
(i) $f[xf(y)]f(y)=f(x+y)$,
(ii) $f(2)=0$ but $f(x)\ne0$ for $0\le x\lt 2$.

Prove or disprove: Given a finite set of points with integer coefficients in the plane, it is possible to color some of these points red and the remaining ones white in such a way that for any straight line $L$ parallel to one of the coordinate axes, the number of red colored points and the number of white colored points on $L$ differ by at most 1.