IMO 1992

Problem 1. Find all integers $a,b,c$, with $1<a<b<c$, such that

$(a-1)(b-1)(c-1)\text{is a divisor of abc−1.}$

Problem 2. Let $\mathbf{R}$, denote the set of all real numbers. Find all functions $f:\mathbf{R}\to \mathbf{R}$, such that

Problem 3. Consider nine points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $n$, such that whenever exactly $n$, edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.

Problem 4. In the plane let $C$, be a circle, $L$, a line tangent to the circle $C,$, and $M$, a point on $L$. Find the locus of all points $P$, with the following property: there exists two points $Q,R$, on $L$, such that $M$, is the midpoint of $QR$, and $C$, is the inscribed circle of triangle $PQR$.

Problem 5. Let $S$, be a finite set of points in three-dimensional space. Let $S_x,S_y,S_z$, be the sets consisting of the orthogonal projections of the points of $S$, onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that

$|S{|}^2\le |S_x|\cdot |S_y|\cdot |S_z|,$

where $|A|$, denotes the number of elements in the finite set $|A|$. (Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane.)

Problem 6. For each positive integer $n,S(n)$, is defined to be the greatest integer such that, for every positive integer $k\le S(n),n^2$, can be written as the sum of $k$, positive squares.

(a) Prove that $S(n)\le n^2-14$, for each $n\ge 4$.

(b) Find an integer $n$, such that $S(n)=n^2-14$.

(c) Prove that there are infintely many integers $n$, such that $S(n)=n^2-14$.