2. The positive integers $x$ and $y$ satisfy the equation $y x^{2}+x y^{2}=70$. What is the value of $x^{4}+y^{4}$ ?
SolUtion
641

We may assume that $x \leq y$ as the expressions $y x^{2}+x y^{2}$ and $x^{4}+y^{4}$ are symmetrical in $x$ and $y$. If $x \geq 4$ (and so $y \geq 4$ ) then $y x^{2}+x y^{2} \geq 2 \times 4^{3}=128>70$. So $x$ must be one of $1,2,3$. When $x=2$ we obtain the equation $4 y+2 y^{2}=70$ which has the integer solution $y=5$. The analogous equations in $y$ obtained for $x=1,3$ do not have integer solutions. Therefore, $x=2, y=5$ and $x^{4}+y^{4}=2^{4}+5^{4}=16+625=641$.