4th Iberoamerican 1989


A1. Find all real solutions to: x + y - z = -1; x² - y² + z² = 1, -x³ + y³ + z³ = -1.
A2. Given positive real numbers x, y, z each less than π/2, show that π/2 + 2 sin x cos y + 2 sin y cos z > sin 2x + sin 2y + sin 2z.
A3. If a, b, c, are the sides of a triangle, show that (a - b)/(a + b) + (b - c)/(b + c) + (c - a)/(a + c) < 1/16.
B1. The incircle of the triangle ABC touches AC at M and BC at N and has center O. AO meets MN at P and BO meets MN at Q. Show that MP·OA = BC·OQ.
B2. The function f on the positive integers satisfies f(1) = 1, f(2n + 1) = f(2n) + 1 and f(2n) = 3 f(n). Find the set of all m such that m = f(n) for some n.
B3. Show that there are infinitely many solutions in positive integers to 2a² - 3a + 1= 3b² + b.

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