IMO 1999

Official IMO 1999 problems  ·  6/6 solved.

Problem 1   solved · 14m15s · Solution →

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 $S$.

Problem 2   solved · 13m57s · Solution →

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 $C$.

Problem 3   solved · 14m45s · Solution →

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 $N$.

Problem 4   solved · 13m43s · Solution →

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 $n^{p-1}$.

Problem 5   solved · 13m54s · Solution →

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 $G_{2}$.

Problem 6   solved · 13m58s · Solution →

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 $x,y$.