IMO 1999

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 for $S$.

Problem 2. Let $n$ be a fixed integer, with $n\ge 2$.

(a) Determine the least constant $C$ such that the inequality

${\sum }_{1\le i<j\le n}{x}_i{x}_j(x_i^2+x_j^2)\le C{\left({\sum }_{1\le i\le n}{x}_i\right)}^4$

holds for all real numbers $x_1,\cdots ,x_n\ge 0$.

(b) For this constant $C$, determine when equality holds.

Problem 3. Consider an $n\times n$ square board, where $n$ is a fixed even positive integer. The board is divided into $n^2$ unit 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. 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. 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. Determine all functions $f:\ast \ast R\ast \ast ⟶\ast \ast R\ast \ast$ such that

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

for all real numbers $x,y$.