1996 USAMO Problems
Contents
Day 1
Problem 1
Prove that the average of the numbers is .
Problem 2
For any nonempty set of real numbers, let denote the sum of the elements of . Given a set of positive integers, consider the collection of all distinct sums as ranges over the nonempty subsets of . Prove that this collection of sums can be partitioned into classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.
Problem 3
Let be a triangle. Prove that there is a line (in the plane of triangle ) such that the intersection of the interior of triangle and the interior of its reflection in has area more than the area of triangle .
Day 2
Problem 4
An -term sequence in which each term is either 0 or 1 is called a binary sequence of length . Let be the number of binary sequences of length containing no three consecutive terms equal to 0, 1, 0 in that order. Let be the number of binary sequences of length that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that for all positive integers .
Problem 5
Let be a triangle, and an interior point such that , , and . Prove that the triangle is isosceles.
Problem 6
Determine (with proof) whether there is a subset of the integers with the following property: for any integer there is exactly one solution of with .
See Also
1996 USAMO (Problems • Resources) | ||
Preceded by 1995 USAMO |
Followed by 1997 USAMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.