First day

Bucharest, Romania - 16th July, 1999

1

A set *S* of points from the plane will be called ** completely
symmetric
** if it has at least three elements and fulfils the condition:
for every distinct points *A,B* from *S*, the perpendicular
bisector of the segment *AB* is an axis of symmetry for *S*.

Prove that if a completely symmetric set is finite then it consists of the vertices of a regular polygon.

2

Let *n> 1* be a fixed integer. Find the least constant *C*
such that the inequality

holds for any

For this constant, characterize the instances of equality.

3

Let *n* be an even positive integer. We say that two different
cells of a *n* x *n* board are ** neighboring ** if they
have a comon side. Find the minimal number of cells on the *n* x
*n* board that must be marked so that any cell (marked or not
marked) has a marked neighboring cell.

Each problem is worth 7 points.

Time: 4 1/2 hours.

Second day

Bucharest, Romania - 17th July, 1999

4

Find all pairs of positive integers (*n,p*) such that *p* is a
prime, *n <2p+1* ,and *x ^{ p}*-1 is a divisor of
(

5

Two circles * C _{}* 1 and

6

Find all the functions *f* from real numbers to real numbers which
satisfy

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

for all real numbersEach problem is worth 7 points.

Time: 4 1/2 hours.