2017 USAJMO Problems
Contents
[hide]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
Consider the equation
![\[\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = (x-y)^7.\]](http://latex.artofproblemsolving.com/5/1/5/5156a3323a4c2b47a971e779fb93769c35814611.png)
(a) Prove that there are infinitely many pairs 
of positive integers satisfying the equation.
(b) Describe all pairs of positive integers satisfying the equation.
Problem 3
() Let 
be an equilateral triangle and let 
be a point on its circumcircle. Let lines 
and 
intersect at 
; let lines 
and 
intersect at 
; and let lines 
and 
intersect at 
. Prove that the area of triangle 
is twice the area of triangle .
Day 2
Problem 4
Problem 5
Problem 6
Let 
be 
distinct points on the unit circle 
other than 
. Each point is colored either red or blue, with exactly 
of them red and of them blue. 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 
traveling 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.
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
| 2017 USAJMO (Problems • Resources) | ||
| Preceded by 2016 USAJMO |
Followed by 2018 USAJMO | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAJMO Problems and Solutions | ||
