2017 USAMO Problems
Contents
Day 1
Note: For any geometry problem whose statement begins with an asterisk (), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.
Problem 1
Prove that there are infinitely many distinct pairs of relatively prime positive integers and such that is divisible by
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: or 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.
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 .
Day 2
Note: For any geometry problem whose statement begins with an asterisk (), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.
Problem 4
Let , , , be distinct points on the unit circle , other than . Each point is colored either red or blue, with exactly red points and blue points. Let , , , be any ordering of the red points. Let be the nearest blue point to traveling counterclockwise around the circle starting from . Then let be the nearest of the remaining blue points to travelling counterclockwise around the circle from , and so on, until we have labeled all of the blue points . Show that the number of counterclockwise arcs of the form that contain the point is independent of the way we chose the ordering of the red points.
Problem 5
Let denote the set of all integers. Find all real numbers such that there exists a labeling of the lattice points with positive integers for which: only finitely many distinct labels occur, and for each label , the distance between any two points labeled is at least .
Problem 6
Find the minimum possible value of given that , , , are nonnegative real numbers such that .
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
2017 USAMO (Problems • Resources) | ||
Preceded by 2016 USAMO |
Followed by 2018 USAMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |