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The 1st United States of America Mathematics Olympiad
1972年第一届美国数学奥林匹克
  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.
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