1982 USAMO Problems
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?
Show that if are positive integers such that for all real with sum , then or .
is a point inside the equilateral triangle . is a point inside . Show that
Show that there is a positive integer such that, for every positive integer , is composite.
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 .
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