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 [i]all[/i] 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
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 .
|1982 USAMO (Problems • Resources)|
|1 • 2 • 3 • 4 • 5|
|All USAMO Problems and Solutions|