Question bank IMO 1999

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 $S$.

Find the least constant $C$ such that for any integer $n \gt 1$ the inequality $$
\sum_{1 \le i \lt j \le n} x_i x_j (x_i^2 + x_j^2)
\le C \left( \sum_{1 \le i \le n} x_i \right)^4
$$ holds for all real numbers $x_1, \dots, x_n \ge 0$. Determine the cases of equality.

Consider a square board of size $n\times n$, where $n$ is a fixed even positive integer. The board is divided into $n^2$ unit squares. Two different squares are adjacent if they have a common side. A total of $N$ unit squares are marked so that every square, marked or unmarked, is adjacent to at least one marked square. Determine the smallest possible value of $N$.

Find all pairs of positive integers $(x,p)$ such that $p$ is a prime, $x^{p-1}$ is a divisor of $(p-1)^{x}+1$, and $x \le 2p$.

Two circles $\Omega_{1}$ and $\Omega_{2}$ touch internally the circle $\Omega$ in $M$ and $N$ and the center of $\Omega_{2}$ is on $\Omega_{1}$. The common chord of the circles $\Omega_{1}$ and $\Omega_{2}$ intersects $\Omega$ in $A$ and $B$ Lines $MA$ and $MB$ intersects $\Omega_{1}$ in $C$ and $D$. Prove that $\Omega_{2}$ is tangent to $CD$.

Find all the functions $f \colon \mathbb{R} \to \mathbb{R}$ such that $$
f(x-f(y))=f(f(y))+xf(y)+f(x)-1
$$ for all $x,y \in \mathbb{R}$.