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

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==Solution==
 
==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}}
 
{{MAA Notice}}

Revision as of 19:24, 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
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