25th International Mathematical Olympiad 1984 Problems & Solutions

A1.  Prove that 0 ≤ yz + zx + xy - 2xyz ≤ 7/27, where x, y and z are non-negative real numbers satisfying x + y + z = 1. A2.  Find one pair of positive integers a, b such that ab(a+b) is not divisible by 7, but (a+b)7 - a7 - b7 is divisible by 77. A3.  Given points O and A in the plane. Every point in the plane is colored with one of a finite number of colors.