1999 IMO Problems

Problems of the 1999 IMO.

Day I

Problem 1

Determine all finite sets $S$ of at least three points in the plane which satisfy the following condition:

For any two distinct points $A$ and $B$ in $S$ , the perpendicular bisector of the line segment $AB$ is an axis of symmetry of $S$ .

Solution

Problem 2

Let $n \geq 2$ be a fixed integer.

(a) Find the least constant $C$ such that for all nonnegative real numbers $x_1, \dots, x_n$ ,

$\sum_{1\leq i<j \leq n} x_ix_j (x_i^2 + x_j^2) \leq C \left( \sum_{i=1}^n x_i \right)^4.$

(b) Determine when equality occurs for this value of $C$ .

Solution

Problem 3

Consider an $n \times n$ square board, where $n$ is a fixed even positive integer. The board is divided into $n^{2}$ units squares. We say that two different squares on the board are adjacent if they have a common side.

$N$ unit squares on the board are marked in such a way that every square (marked or unmarked) on the board is adjacent to at least one marked square.

Determine the smallest possible value of $N$ .

Solution

Day II

Problem 4

Determine all pairs $(n,p)$ of positive integers such that

$p$ is a prime, $n$ not exceeded $2p$ , and $(p-1)^{n}+1$ is divisible by $n^{p-1}$ .

Solution

Problem 5

Two circles $G_{1}$ and $G_{2}$ are contained inside the circle $G$ , and are tangent to $G$ at the distinct points $M$ and $N$ , respectively. $G_{1}$ passes through the center of $G_{2}$ . The line passing through the two points of intersection of $G_{1}$ and $G_{2}$ meets $G$ at $A$ and $B$ . The lines $MA$ and $MB$ meet $G_{1}$ at $C$ and $D$ , respectively.

Prove that $CD$ is tangent to $G_{2}$ .

Solution

Problem 6

Determine all functions $f:\Bbb{R}\to \Bbb{R}$ such that

$f(x-f(y))=f(f(y))+xf(y)+f(x)-1$

for all real numbers $x,y$ .

Solution

1999 IMO (Problems) • Resources
Preceded by 1998 IMO	1 • 2 • 3 • 4 • 5 • 6	Followed by 2000 IMO
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1999 IMO Problems

Contents

Day I

Problem 1

Problem 2

Problem 3

Day II

Problem 4

Problem 5

Problem 6

See Also