Problem Set Workbook
Access the downloadable workbook for 2010 USAMO Day 1 problems here.
Discussion Forum
Engage in discussion about the 2010 USAMO Day 1 math contest by visiting [Random Math USAMO Day 1 2010 Forum](https://forums.randommath.com/c/tournaments/MAA/2010-usamo-day 1)
Individual Problems and Solutions
For problems and detailed solutions to each of the 2010 USAMO Day 1 problems, please refer below:
Problem 1: Let be a convex pentagon inscribed in a semicircle of diameter . Denote by the feet of the perpendiculars from onto lines , respectively. Prove that the acute angle formed by lines and is half the size of , where is the midpoint of segment .
Solution:
Problem 2: There are students standing in a circle, one behind the other. The students have heights . If a student with height is standing directly behind a student with height or less, the two students are permitted to switch places. Prove that it is not possible to make more than such switches before reaching a position in which no further switches are possible.
Solution:
Problem 3: The 2010 positive numbers satisfy the inequality for all distinct indices . Determine, with proof, the largest possible value of the product .
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions