11th International Mathematical Olympiad 1969 Problems & Solutions

A1.  Prove that there are infinitely many positive integers m, such that n4 + m is not prime for any positive integer n. A2.  Let f(x) = cos(a1 + x) + 1/2 cos(a2 + x) + 1/4 cos(a3 + x) + ... + 1/2n-1 cos(an + x), where ai are real constants and x is a real variable. If f(x1) = f(x2) = 0, prove that x1 - x2 is a multiple of π. A3.  For each of k = 1, 2, 3, 4, 5 find