11th USAMO 1982


1. A graph has 1982 points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to 1981 points?
2. Show that if m, n are positive integers such that (x^m+n + y^m+n + z^m+n)/(m+n) = (x^m + y^m + z^m)/m (xⁿ + yⁿ + zⁿ)/n for all real x, y, z with sum 0, then {m, n} = {2, 3} or {2, 5}.
3. D is a point inside the equilateral triangle ABC. E is a point inside DBC. Show that area DBC/(perimeter DBC)² > area EBC/(perimeter EBC)².
4. Show that there is a positive integer k such that, for every positive integer n, k 2ⁿ + 1 is composite.
5. O is the center of a sphere S. Points A, B, C are inside S, OA is perpendicular to AB and AC, and there are two spheres through A, B, and C which touch S. Show that the sum of their radii equals the radius of S.

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