ΒΆ 2009 USAMO Day 1 Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2009 USAMO Day 1 problems here.
Discussion Forum
Engage in discussion about the 2009 USAMO Day 1 math contest by visiting [Random Math USAMO Day 1 2009 Forum](https://forums.randommath.com/c/tournaments/MAA/2009-usamo-day 1)
Individual Problems and Solutions
For problems and detailed solutions to each of the 2009 USAMO Day 1 problems, please refer below:
Problem 1: Given circles and intersecting at points and , let be a line through the center of intersecting at points and and let be a line through the center of intersecting at points and . Prove that if and lie on a circle then the center of this circle lies on line .
Solution:
Problem 2: Let be a positive integer. Determine the size of the largest subset of
which does not contain three elements (not necessarily distinct) satisfying 0 .
Solution:
Problem 3: We define a chessboard polygon to be a polygon whose edges are situated along lines of the form or , where and are integers. These lines divide the interior into unit squares, which are shaded alternately grey and white so that adjacent squares have different colors. To tile a chessboard polygon by dominoes is to exactly cover the polygon by non-overlapping rectangles. Finally, a tasteful tiling is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a rectangle are shown; the first one is tasteful, while the second is not, due to the vertical dominoes in the upper right corner.
- Prove that if a chessboard polygon can be tiled by dominoes, then it can be done so tastefully.
- Prove that such a tasteful tiling is unique.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions