1st Iberoamerican 1985


A1. Find all integer solutions to: a + b + c = 24, a² + b² + c² = 210, abc = 440.
A2. P is a point inside the equilateral triangle ABC such that PA = 5, PB = 7, PC = 8. Find AB.
A3. Find the roots r₁, r₂, r₃, r₄ of the equation 4x⁴ - ax³ + bx² - cx + 5 = 0, given that they are positive reals satisfying r₁/2 + r₂/4 + r₃/5 + r₄/8 = 1.
B1. The reals x, y, z satisfy x ≠ 1, y ≠ 1, x ≠ y, and (yz - x²)/(1 - x) = (xz - y²)/(1 - y). Show that (yx - x²)/(1 - x) = x + y + z.
B2. The function f(n) is defined on the positive integers and takes non-negative integer values. It satisfies (1) f(mn) = f(m) + f(n), (2) f(n) = 0 if the last digit of n is 3, (3) f(10) = 0. Find f(1985).
B3. O is the circumcenter of the triangle ABC. The lines AO, BO, CO meet the opposite sides at D, E, F respectively. Show that 1/AD + 1/BE + 1/CF = 2/AO.

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