9th USAMO 1980


1. A balance has unequal arms and pans of unequal weight. It is used to weigh three objects. The first object balances against a weight A, when placed in the left pan and against a weight a, when placed in the right pan. The corresponding weights for the second object are B and b. The third object balances against a weight C, when placed in the left pan. What is its true weight?
2. Find the maximum possible number of three term arithmetic progressions in a monotone sequence of n distinct reals.
3. A + B + C is an integral multiple of π. x, y, z are real numbers. If x sin A + y sin B + z sin C = x² sin 2A + y² sin 2B + z² sin 2C = 0, show that xⁿ sin nA + yⁿ sin nB + zⁿ sin nC = 0 for any positive integer n.
4. The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular.
5. If x, y, z are reals such that 0 ≤ x, y, z ≤ 1, show that x/(y + z + 1) + y/(z + x + 1) + z/(x + y + 1) ≤ 1 - (1 - x)(1 - y)(1 - z).

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