2017 USAJMO Problems/Problem 1
Contents
Problem
Prove that there are infinitely many distinct pairs of relatively prime integers
and
such that
is divisible by
.
Solution 1
Let and
. We see that
. Therefore, we have
, as desired.
(Credits to laegolas)
Solution 2
Let be any odd number above 1. We have
Since
is even,
This means that
and since x is odd,
or
This means for any odd x, the ordered triple
satisfies the condition. Since there are infinitely many values of
possible, there are infinitely many ordered triples, as desired.
07:28
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
See also
2017 USAJMO (Problems • Resources) | ||
First Problem | Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |