1992 IMO Problems

Problems of the 1992 IMO.

Day I

Problem 1

Find all integers $a$ , $b$ , $c$ satisfying $1 < a < b < c$ such that $(a - 1)(b -1)(c - 1)$ is a divisor of $abc - 1$ .

Solution

Problem 2

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

$f\left( x^{2}+f(y) \right)= y+(f(x))^{2} \hspace{0.5cm} \forall x,y \in \mathbb{R}$

Solution

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.

Solution

Day II

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 $PQR$ .

Solution

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)

Solution

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 infinitely many integers $n$ such that $S(n)=n^{2}-14$ .

Solution

1992 IMO (Problems) • Resources
Preceded by 1991 IMO	1 • 2 • 3 • 4 • 5 • 6	Followed by 1993 IMO
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1992 IMO Problems

Contents

Day I

Problem 1

Problem 2

Problem 3

Day II

Problem 4

Problem 5

Problem 6