14th BALKAN MATHEMATICAL OLYMPIAD
Kalampaka, Greece, April 29, 1997

Problem 1

Let O be an interior point of a convex quadrilateral ABCD satisfaying
OA2 + OB2 + OC2 + OD2 = 2 S(ABCD)
where S(ABCD) denotes the area of ABCD. Show that ABCD is a square with center O.

Problem 2

Let A = {A1, A2, ... , Ak} (k > 1), be a collection of subsets of an n-set S such that for any x, y S there is Ai A such that x Ai and y A \ Ai or x A \ Ai and y Ai. Show that k [log2 n].

Problem 3

Three circles G, C1 and C2 are given in the plane. C1 and C2 touch G internally at points B and C, respectively. Moreover, C1 and C2 touch each other externally at a point D. Let A be one of the points in which the common tangent of C1 and C2 intersects G. Denote by M the second point of intersection of the line AB and the circle C1 and by N the second point of intersection of the line AC and the circle C2. Further denote by K and L the second points of intersection of the line BC with C1 and C2, 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.