English version


Bucharest, July 16, 1999

Time allowed 4 hours 30 min.
Each problem is worth 7 points
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.