2005 USAMO Problems
Contents
Day 1
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.
Problem 2
Prove that the system has no solutions in integers , , and .
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.
Day 2
Problem 4
This problem needs a solution. If you have a solution for it, please help us out by adding it.
Problem 5
This problem needs a solution. If you have a solution for it, please help us out by adding it.
Problem 6
This problem needs a solution. If you have a solution for it, please help us out by adding it.
Resources
- USAMO Problems and Solutions
- 2005 USAMO Problems
- 2005 USAMO Solutions
- USAMO Problems on the Resources page
2005 USAMO (Problems • Resources) | ||
Preceded by 2004 USAMO |
Followed by 2006 USAMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |