Problem Set Workbook
Access the downloadable workbook for 2013 USAMO Day 1 problems here.
Discussion Forum
Engage in discussion about the 2013 USAMO Day 1 math contest by visiting [Random Math USAMO Day 1 2013 Forum](https://forums.randommath.com/c/tournaments/MAA/2013-usamo-day 1)
Individual Problems and Solutions
For problems and detailed solutions to each of the 2013 USAMO Day 1 problems, please refer below:
Problem 1: In triangle , points lie on sides , respectively. Let denote the circumcircles of triangles , respectively. Given the fact that segment intersects again at respectively, prove that .
Solution:
Problem 2: For a positive integer plot equally spaced points around a circle. Label one of them , and place a marker at . One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of distinct moves available; two from each point. Let count the number the number of ways to advance around the circle exactly twice, beginning and ending at , without repeating a move. Prove that for all .
Solution:
Problem 3: Let be a positive integer. There are marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing marks. Initially, each mark has the black side up. An operation is to choose a line parallel to one of the sides of the triangle, and flipping all the marks on that line. A configuration is called admissible if it can be obtained from the initial configuration by performing a finite number of operations. For each admissible configuration , let denote the smallest number of operations required to obtain from the initial configuration. Find the maximum value of , where varies over all admissible configurations.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions