1972 USAMO Problems
Problems from the 1972 USAMO.
Problem 1
The symbols and denote the greatest common divisor and least common multiple, respectively, of the positive integers . For example, and . Prove that
Problem 2
A given tetrahedron is isosceles, that is, . Show that the faces of the tetrahedron are acute-angled triangles.
Problem 3
A random number selector can only select one of the nine integers 1, 2, ..., 9, and it makes these selections with equal probability. Determine the probability that after selections (), the product of the numbers selected will be divisible by 10.
Problem 4
Let denote a non-negative rational number. Determine a fixed set of integers , such that for every choice of ,
Problem 5
A given convex pentagon has the property that the area of each of the five triangles , , , , and is unity. Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.
See Also
1972 USAMO (Problems • Resources) | ||
Preceded by First USAMO |
Followed by 1973 USAMO | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.