2019 USAJMO Problems/Problem 1
Revision as of 20:54, 18 April 2019 by Brendanb4321 (talk | contribs) (Created page with "==Problem== There are <math>a+b</math> bowls arranged in a row, number <math>1</math> through <math>a+b</math>, where <math>a</math> and <math>b</math> are given positive inte...")
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
These problems are copyrighted © by the Mathematical Association of America, as part of the 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 | ||
