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
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 |