2019 USAJMO 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
There are 
bowls arranged in a row, numbered 
through , where 
and 
are given positive integers. Initially, each of the first bowls contains an apple, and each of the last bowls contains a pear.
A legal move consists of moving an apple from bowl 
to bowl 
and a pear from bowl 
to bowl 
, provided that the difference 
is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first bowls each containing a pear and the last bowls each containing an apple. Show that this is possible if and only if the product 
is even.
Problem 2
Let 
be the set of all integers. Find all pairs of integers 
for which there exist functions 
and 
satisfying ![\[f(g(x))=x+a\quad\text{and}\quad g(f(x))=x+b\]](http://latex.artofproblemsolving.com/d/6/9/d69627a699096287d1de16796019aacb2c1c9e9a.png)
for all integers 
.
Problem 3
Let 
be a cyclic quadrilateral satisfying 
. The diagonals of intersect at 
. Let 
be a point on side 
satisfying 
. Show that line 
bisects 
.
Day 2
Problem 4
Let 
be a triangle with 
obtuse. The 
-excircle is a circle in the exterior of 
that is tangent to side 
of the triangle and tangent to the extensions of the other two sides. Let 
be the feet of the altitudes from 
and 
to lines 
and 
, respectively. Can line 
be tangent to the -excircle?
Problem 5
Problem 6
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
| 2019 USAJMO (Problems • Resources) | ||
| Preceded by 2018 USAJMO |
Followed by 2020 USAJMO | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAJMO Problems and Solutions | ||
