International Competitions 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 $S$.

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

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

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

holds for all real numbers $x_1, \ldots, x_n \geq 0$.

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

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 $N$.

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 $n^{p-1}$.

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 $G_2$.

Determine all functions $f: \mathbf{R} \longrightarrow \mathbf{R}$ such that

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

for all real numbers $x, y$.