2017 USAJMO Problems/Problem 2
Problem:
Consider the equation
(a) Prove that there are infinitely many pairs of positive integers satisfying the equation.
(b) Describe all pairs of positive integers satisfying the equation.
Solution 1 (and motivation)
We have , which can be expressed as . At this point, we think of substitution. A substitution of form is slightly derailed by the leftover x and y terms, so instead, seeing the xy in front, we substitute . This leaves us with , so . Expanding yields . Rearranging, we have . To satisfy this equation in integers, must obviously be a power, and further inspection shows that it must also be odd. Also, since it is a square and all odd squares are 1 mod 8, every odd sixth power gives a solution. Since the problem asks for positive integers, the pair does not work. We go to the next highest odd power, or . In this case, , so the LHS is , so . Using the original substitution yields as the first solution. We have shown part a by showing that there are an infinite number of positive integer solutions for , which can then be manipulated into solutions for . To solve part b, we look back at the original method of generating solutions. Define and to be the pair representing the nth solution. Since the nth odd number is , . It follows that . From our original substitution, . {plshalp} The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
See also
2017 USAJMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |