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

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(Problem:)
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==Problem:==
 
==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>.  
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Consider the equation
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<cmath>\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = (x-y)^7.</cmath>
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(a) Prove that there are infinitely many pairs <math>(x,y)</math> of positive integers satisfying the equation.
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(b) Describe all pairs <math>(x,y)</math> of positive integers satisfying the equation.
  
 
==Solution==
 
==Solution==

Revision as of 19:12, 19 April 2017

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

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See also

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