14th Balkan 1997


A1. ABCD is a convex quadrilateral. X is a point inside it. XA² + XB² + XC² + XD² is twice the area of the quadrilateral. Show that it is a square and that X is its center.
A2. A collection of m subsets of X = {1, 2, ... , n} has the property that given any two elements of X we can find a subset in the collection which contains just one of the two. Prove that n ≤ 2^m.
A3. Two circles C and C' lying outside each other touch at T. They lie inside a third circle and touch it at X and X' respectively. Their common tangent at T intersects the third circle at S. SX meets C again at P and XX' meets C again at Q. SX' meets C' again at U and XX' meets C' again at V. Prove that the lines ST, PQ and UV are concurrent.
A4. Find all real-valued functions on the reals which satisfy f( xf(x) + f(y) ) = f(x)² + y for all x, y.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.