# Difference between revisions of "2017 USAJMO Problems/Problem 2"

## Problem:

Consider the equation $$\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = (x-y)^7.$$

(a) Prove that there are infinitely many pairs $(x,y)$ of positive integers satisfying the equation.

(b) Describe all pairs $(x,y)$ of positive integers satisfying the equation.

## Solution

Part a: Let $y = an$ and $x = a(n + 1)$. Substituting, we have $$a^7 = a^6 \left(3(n+1)^3 + (n+1)n^2 \right) \left(3n^3 + n(n+1)^2 \right).$$ Therefore, we have $$a = \left(3(n+1)^3 + (n+1)n^2 \right) \left(3n^3 + n(n+1)^2 \right),$$ which implies that there is a solution for every positive integer $n$