Consider the equation $(3x^3 + xy^2)(x^2y + 3y^3) = (x - y)^7$

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.

## USA(J)MO 2017 #2

### USA(J)MO 2017 #2

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### Re: USA(J)MO 2017 #2

Substitute $a=x+y$ and $b=x-y$ and after some simplification, we get

$a^6 = b^6(4b+1)$

So, $4b+1=(2n+1)^6$

Here we'll find a value of $b$ in terms of $n$. Then $a=(2n+1)b$, here we'll input the value of $b$ and get a value of $a$ in terms of $n$. $x=\frac{a+b}{2}$ and $y=\frac{a-b}{2}$ .

$a^6 = b^6(4b+1)$

So, $4b+1=(2n+1)^6$

Here we'll find a value of $b$ in terms of $n$. Then $a=(2n+1)b$, here we'll input the value of $b$ and get a value of $a$ in terms of $n$. $x=\frac{a+b}{2}$ and $y=\frac{a-b}{2}$ .

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