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

Revision as of 18:12, 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 1: / Line 1: @@
 ==Problem:==
-Prove that there are infinitely many distinct pairs <math>(a,b)</math> of relatively prime positive integers <math>a > 1</math> and <math>b > 1</math> such that <math>a^b + b^a</math> is divisible by <math>a + b</math>.
+Consider the equation
+<cmath>\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = (x-y)^7.</cmath>
+	(a) Prove that there are infinitely many pairs <math>(x,y)</math> of positive integers satisfying the equation.
+	(b) Describe all pairs <math>(x,y)</math> of positive integers satisfying the equation.
 ==Solution==

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