1979 USAMO Problems
Problems from the 1979 USAMO.
Determine all non-negative integral solutions if any, apart from permutations, of the Diophantine Equation .
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).
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 .
lies between the rays and . Find on and on collinear with so that is as large as possible.
Let be distinct subsets of with . Prove that for some pair . Note that , or, alternatively, .
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