English version

FIRST DAY

Bucharest, July 16, 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 .

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

$$

holds for all real numbers $$ .

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

Problem 3.

Consider an $n\; x\; n$ square board, where $n$ is a fixed even positive integer. The board is divided into $n2$ 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 $N$.