2019 USAJMO Problems/Problem 1
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Problem
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.
Solution
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
See also
| 2019 USAJMO (Problems • Resources) | ||
| First Problem | Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAJMO Problems and Solutions | ||
