Problem Set Workbook
Access the downloadable workbook for 2011 USAMO Day 1 problems here.
Discussion Forum
Engage in discussion about the 2011 USAMO Day 1 math contest by visiting [Random Math USAMO Day 1 2011 Forum](https://forums.randommath.com/c/tournaments/MAA/2011-usamo-day 1)
Individual Problems and Solutions
For problems and detailed solutions to each of the 2011 USAMO Day 1 problems, please refer below:
Problem 1: Let be positive real numbers such that . Prove that
Solution:
Problem 2: An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer from each of the integers at two neighboring vertices and adding to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0 . Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won.
Solution:
Problem 3: In hexagon , which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy , and . Furthermore , and . Prove that diagonals , and are concurrent.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions