Problem Set Workbook
Access the downloadable workbook for 2012 USAMO Day 1 problems here.
Discussion Forum
Engage in discussion about the 2012 USAMO Day 1 math contest by visiting [Random Math USAMO Day 1 2012 Forum](https://forums.randommath.com/c/tournaments/MAA/2012-usamo-day 1)
Individual Problems and Solutions
For problems and detailed solutions to each of the 2012 USAMO Day 1 problems, please refer below:
Problem 1: Find all integers such that among any positive real numbers with
there exist three that are the side lengths of an acute triangle.
Solution:
Problem 2: A circle is divided into 432 congruent arcs by 432 points. The points are colored in four colors such that some 108 points are colored Red, some 108 points are colored Green, some 108 points are colored Blue, and the remaining 108 points are colored Yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.
Solution:
Problem 3: Determine which integers have the property that there exists an infinite sequence of nonzero integers such that the equality
holds for every positive integer .
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions