14th Iberoamerican Mathematical Olympiad Problems 1999

A1.  Find all positive integers n < 1000 such that the cube of the sum of the digits of n equals n2. A2.  Given two circles C and C' we say that C bisects C' if their common chord is a diameter of C'. Show that for any two circles which are not concentric, there are infinitely many circles which bisect them both. Find the locus of the centers of the