1982 USAMO Problems

Problems from the 1982 USAMO.

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?

Solution

Problem 2

Show that if are positive integers such that $\frac{\left(x^{m+n} + y^{m+n} + z^{m+n}\right)}{m+n}=\left(\frac{x^m + y^m + z^m}{m}\right) \left(\dfrac{x^n + y^n + z^n}{n}}\right)$ (Error compiling LaTeX. Unknown error_msg) for all real with sum , then or .

Solution

Problem 3

is a point inside the equilateral triangle . is a point inside . Show that

Solution

Problem 4

Show that there is a positive integer such that, for every positive integer , is composite.

Solution

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 .

Solution

See Also

1982 USAMO (Problems • Resources)
Preceded by 1981 USAMO	Followed by 1983 USAMO
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions