Individual Problems and Solutions
For problems and detailed solutions to each of the 2003 USAMO problems, please refer below:
Problem 4: Let be a triangle. A circle passing through and intersects segments and at and , respectively. Lines and intersect at while lines and intersect at . Prove that if and only if .
Solution:
Problem 5: Let be positive real numbers. Prove that
Solution:
Problem 6: A positive integer is written at each vertex of a regular hexagon so that the sum of all numbers written is . Bert makes a sequence of moves of the following form: Bert picks a vertex and replaces the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can always make a sequence of moves ending at the position with all six numbers equal to zero.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions