6th APMO 1994


A1. Find all real-valued functions f on the reals such that (1) f(1) = 1, (2) f(-1) = -1, (3) f(x) ≤ f(0) for 0 < x < 1, (4) f(x + y) ≥ f(x) + f(y) for all x, y, (5) f(x + y) ≤ f(x) + f(y) + 1 for all x, y.
A2. ABC is a triangle and A, B, C are not collinear. Prove that the distance between the orthocenter and the circumcenter is less than three times the circumradius.
A3. Find all positive integers n such that n = a² + b², where a and b are relatively prime positive integers, and every prime not exceeding √n divides ab.
A4. Can you find infinitely many points in the plane such that the distance between any two is rational and no three are collinear?
A5. Prove that for any n > 1 there is either a power of 10 with n digits in base 2 or a power of 10 with n digits in base 5, but not both.

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