Problem Set Workbook
Access the downloadable workbook for 2017 USAMO Day 1 problems here.
Discussion Forum
Engage in discussion about the 2017 USAMO Day 1 math contest by visiting [Random Math USAMO Day 1 2017 Forum](https://forums.randommath.com/c/tournaments/MAA/2017-usamo-day 1)
Individual Problems and Solutions
For problems and detailed solutions to each of the 2017 USAMO Day 1 problems, please refer below:
Problem 1: Prove that there are infinitely many distinct pairs of relatively prime integers and such that is divisible by .
Solution:
Problem 2: Let be a collection of positive integers, not necessarily distinct. For any sequence of integers and any permutation of , define an -inversion of to be a pair of entries with for which one of the following conditions holds:
Show that, for any two sequences of integers and , and for any positive integer , the number of permutations of having exactly -inversions is equal to the number of permutations of having exactly -inversions.
Solution:
Problem 3: Let be a scalene triangle with circumcircle and incenter . Ray meets at and meets again at ; the circle with diameter cuts again at . Lines and meet at , and is the midpoint of . The circumcircles of and intersect at points and . Prove that passes through the midpoint of either or .
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions