International Competitions 1992

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

$(a - 1)(b - 1)(c - 1)$ is a divisor of $abc - 1$.

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

$$f \left(x^2 + f(y)\right) = y + \left(f(x)\right)^2 \quad \text{for all } x, y \in R.$$

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.

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 $PQR$.

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

$$|S|^2 \leq |S_x| \cdot |S_y| \cdot |S_z|,$$

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.)

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

(a) Prove that $S(n) \leq n^2 - 14$ for each $n \geq 4$.

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

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