English version

Bucharest, July 16, 1999

Time allowed 4 hours 30 min. |

Each problem is worth 7 points |

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 *$N$*.