1982 USAMO Problems
Problems from the 1982 USAMO.
In a party with persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else?
Let with real. It is known that if ,
for , or . Determine all other pairs of integers if any, so that holds for all real numbers such that .
If a point is in the interior of an equilateral triangle and point is in the interior of , prove that
where the isoperimetric quotient of a figure is defined by
Prove that there exists a positive integer such that is composite for every positive integer .
, and are three interior points of a sphere such that and are perpendicular to the diameter of through , and so that two spheres can be constructed through , , and which are both tangent to . Prove that the sum of their radii is equal to the radius of .
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