1979 USAMO Problems
Problems from the 1979 USAMO.
Problem 1
Determine all non-negative integral solutions if any, apart from permutations, of the Diophantine Equation .
Problem 2
is the north pole. and are points on a great circle through equidistant from . is a point on the equator. Show that the great circle through and bisects the angle in the spherical triangle (a spherical triangle has great circle arcs as sides).
Problem 3
is an arbitrary sequence of positive integers. A member of the sequence is picked at random. Its value is . Another member is picked at random, independently of the first. Its value is . Then a third value, . Show that the probability that is divisible by is at least .
Problem 4
lies between the rays and . Find on and on collinear with so that is as large as possible.
Problem 5
Let be distinct subsets of with . Prove that for some pair . Note that , or, alternatively, .
See Also
1979 USAMO (Problems • Resources) | ||
Preceded by 1978 USAMO |
Followed by 1980 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.