2019 USAJMO
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, number
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
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
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 |