2017 USAJMO Problems/Problem 1
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 mathmaster2012)
Solution 2
Let be odd where . We have so This means that and since x is odd, or as desired.
Solution 3
Because problems such as this usually are related to expressions along the lines of , it's tempting to try these. After a few cases, we see that is convenient due to the repeated occurrence of when squared and added. We rewrite the given expressions as: After repeatedly factoring the initial equation,we can get: Expanding each of the squares, we can compute each product independently then sum them: Now we place the values back into the expression: Plugging any positive integer value for into yields a valid solution, because there is an infinite number of positive integers, there is an infinite number of distinct pairs . -fatant
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 |