9th International Mathematical Olympiad 1967 Problems & Solutions

A1.  The parallelogram ABCD has AB = a, AD = 1, angle BAD = A, and the triangle ABD has all angles acute. Prove that circles radius 1 and center A, B, C, D cover the parallelogram iff             a ≤ cos A + √3 sin A. A2.  Prove that a tetrahedron with just one edge length greater than 1 has volume at most 1/8. A3.  Let k, m, n be natural numbers such that m + k