IMO 1992 - Official contest problems

Find all integer triples $(p,q,r)$ such that $1\lt p\lt q\lt r$ and $(p-1)(q-1)(r-1)$ is a divisor of $(pqr-1)$.

Find all functions $f:\mathbb R\to\mathbb R$ such that
$$f(x^2+f(y))=y+f(x)^2$$
for all $x,y\in\mathbb R$.

Given nine points in space, no four of which are coplanar, find the minimal natural number $n$ such that for any coloring with red or blue of $n$ edges drawn between these nine points there always exists a triangle having all edges of the same color.

In the plane, let there be given a circle $C$, a line $l$ tangent to $C$, and a point $M$ on $l$. Find the locus of points $P$ that has the following property: There exist two points $Q$ and $R$ on $l$ such that $M$ is the midpoint of $QR$ and $C$ is the incircle of $PQR$.

Let $V$ be a finite subset of Euclidean space consisting of points $(x,y,z)$ with integer coordinates. Let $S_1,S_2,S_3$ be the projections of $V$ onto the $yz,xz,xy$ planes, respectively. Prove that
$$|V|^2\le |S_1||S_2||S_3|$$
($|X|$ denotes the number of elements of $X$).

For each positive integer $n$, denote by $s(n)$ the greatest integer such that for all positive integer $k\le s(n)$, $n^2$ can be expressed as a sum of squares of $k$ positive integers.
(a) Prove that $s(n)\le n^2-14$ for all $n\ge4$.
(b) Find a number $n$ such that $s(n)=n^2-14$.
(c) Prove that there exist infinitely many positive integers $n$ such that $s(n)=n^2-14$.