Problem Set Workbook
Access the downloadable workbook for 2005 USAMO Day 1 problems here.
Discussion Forum
Engage in discussion about the 2005 USAMO Day 1 math contest by visiting [Random Math USAMO Day 1 2005 Forum](https://forums.randommath.com/c/tournaments/MAA/2005-usamo-day 1)
Individual Problems and Solutions
For problems and detailed solutions to each of the 2005 USAMO Day 1 problems, please refer below:
Problem 1: Determine all composite positive integers for which it is possible to arrange all divisors of that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.
Solution:
Problem 2: Prove that the system
has no solutions in integers , and .
Solution:
Problem 3: Let be an acute-angled triangle, and let and be two points on side . Construct point in such a way that convex quadrilateral is cyclic, , and and lie on opposite sides of line . Construct point in such a way that convex quadrilateral is cyclic, , and and lie on opposite sides of line . Prove that points , and lie on a circle.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions