1st USAMO 1972


1. Let (a, b, ... , k) denote the greatest common divisor of the integers a, b, ... k and [a, b, ... , k] denote their least common multiple. Show that for any positive integers a, b, c we have (a, b, c)² [a, b] [b, c] [c, a] = [a, b, c]² (a, b) (b, c) (c, a).
2. A tetrahedron has opposite sides equal. Show that all faces are acute-angled.
3. n digits, none of them 0, are randomly (and independently) generated, find the probability that their product is divisible by 10.
4. Let k be the real cube root of 2. Find integers A, B, C, a, b, c such that \| (Ax² + Bx + C)/(ax² + bx + c) - k \| < \| x - k \| for all non-negative rational x.
5. A pentagon is such that each triangle formed by three adjacent vertices has area 1. Find its area, but show that there are infinitely many incongruent pentagons with this property.

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