### 14^{th} BALKAN MATHEMATICAL OLYMPIAD
Kalampaka, Greece, April 29, 1997

### Problem 1

Let *O* be an interior point of a convex quadrilateral *ABCD* satisfaying
*OA*^{2} + OB^{2} + OC^{2} +
OD^{2} = 2 *S(ABCD)*
where *S(ABCD)* denotes the area of *ABCD*. Show that
*ABCD* is a square with center *O*.
### Problem 2

Let *A* = {*A*_{1}, *A*_{2}, ... ,
*A*_{k}} (*k* > 1), be a collection of subsets of
an
*n*-set *S* such that for any *x*, *y* Î *S* there is *A*_{i}
Î *A* such that *x* Î *A*_{i} and
*y* Î *A* \ *A*_{i} or
*x* Î *A* \ *A*_{i}
and *y* Î *A*_{i}. Show that
*k* ³
[log_{2} *n*].
### Problem 3

Three circles G, *C*_{1} and
*C*_{2} are given in the plane. *C*_{1}
and *C*_{2} touch G internally at
points *B* and *C*, respectively.
Moreover, *C*_{1} and *C*_{2} touch each other
externally at a point *D*. Let *A*
be one of the points in which the common tangent of *C*_{1}
and *C*_{2} intersects G.
Denote by *M* the second point of intersection of the line *AB*
and the circle *C*_{1} and by *N* the second
point of intersection of the line *AC* and the circle *C*_{2}. Further denote by *K* and *L*
the second points of intersection of the line *BC* with
*C*_{1} and *C*_{2}, respectively.
Show that the lines *AD, MK* and *NL* are concurrent.
### Problem 4

Find all functions *f* : **R** ® **R** such that

*f (xf(x) + f(y)) = (f (x))*^{2} + *y*
holds for all *x* and *y*.