Difference between revisions of "2017 USAJMO Problems/Problem 2"
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| − | + | 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 18:15, 19 April 2017
Problem:
Consider the equation
![\[\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = (x-y)^7.\]](http://latex.artofproblemsolving.com/5/1/5/5156a3323a4c2b47a971e779fb93769c35814611.png)
(a) Prove that there are infinitely many pairs 
of positive integers satisfying the equation.
(b) Describe all pairs of positive integers satisfying the equation.
Solution
Part a: Let 
and 
. Substituting, we have
![\[a^7 = a^6 \left(3(n+1)^3 + (n+1)n^2 \right) \left(3n^3 + n(n+1)^2 \right).\]](http://latex.artofproblemsolving.com/0/7/1/071d930c472516a7fd41720ca56a9217918f16bd.png)
Therefore, we have
![\[a = \left(3(n+1)^3 + (n+1)n^2 \right) \left(3n^3 + n(n+1)^2 \right),\]](http://latex.artofproblemsolving.com/a/b/f/abfef52487830f548f4ab29d35a7a5bf31a045d4.png)
which implies that there is a solution for every positive integer 
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
See also
| 2017 USAJMO (Problems • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAJMO Problems and Solutions | ||
