Individual Problems and Solutions
For problems and detailed solutions to each of the 2002 USAMO problems, please refer below:
Problem 1: Let be a set with elements, and let be an integer with . Prove that it is possible to color every subset of either black or white so that the following conditions hold:
a. the union of any two white subsets is white;
b. the union of any two black subsets is black;
c. there are exactly white subsets.
Solution:
Problem 2: Let be a triangle such that
where and denote its semiperimeter and its inradius, respectively. Prove that triangle is similar to a triangle whose side lengths are all positive integers with no common divisor and determine these integers.
Solution:
Problem 3: Prove that any monic polynomial (a polynomial with leading coefficient ) of degree with real coefficients is the average of two monic polynomials of degree with real roots.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions