Problem Set Workbook
Access the downloadable workbook for 2004 USAMO problems here.
Discussion Forum
Engage in discussion about the 2004 USAMO math contest by visiting Random Math USAMO 2004 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2004 USAMO problems, please refer below:
Problem 4: Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing in the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number in that row is colored black. Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can't. (If two squares share a vertex, Alice can draw a line from one to the other that stays in those two squares.) Find, with proof, a winning strategy for one of the players.
Solution:
Problem 5: Let and be positive real numbers. Prove that
Solution:
Problem 6: A circle is inscribed in a quadrilateral . Let be the center of . Suppose that
Prove that is an isosceles trapezoid.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions