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

Revision as of 19:24, 19 April 2017

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.

@@ Line 8: / Line 8: @@
 ==Solution==
-Part a: Let <math>y = an</math> and <math>x = a(n + 1)</math>. Substituting, we have
-<cmath>a^7 = a^6 \left(3(n+1)^3 + (n+1)n^2 \right) \left(3n^3 + n(n+1)^2 \right).</cmath>
-Therefore, we have
-<cmath>a = \left(3(n+1)^3 + (n+1)n^2 \right) \left(3n^3 + n(n+1)^2 \right),</cmath>
-which implies that there is a solution for every positive integer <math>n</math>.
 {{MAA Notice}}

2017 USAJMO (Problems • Resources)
Preceded by Problem 1	Followed by Problem 3
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions