1982 USAMO Problems
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[hide]Problem 1
A graph has points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to points?
Problem 2
Show that if are positive integers such that for all real with sum , then or .
Problem 3
is a point inside the equilateral triangle . is a point inside . Show that
Problem 4
Show that there is a positive integer such that, for every positive integer , is composite.
Problem 5
is the center of a sphere . Points are inside , is perpendicular to and , and there are two spheres through , and which touch . Show that the sum of their radii equals the radius of .