
Problem 10a.3. Find all positive integers $x$ and $y$ such that

$$
\frac{x^{3}+y^{3}-x^{2} y^{2}}{(x+y)^{2}}
$$

is a nonnegative integer.


Solution: Let $x$ and $y$ be positive integers such that

$$
z=\frac{x^{3}+y^{3}-x^{2} y^{2}}{(x+y)^{2}}
$$

is a nonnegative integer. Substitute $a=x+y$ and $b=x y$ and write the expression in the form $b^{2}+3 a b-a^{2}(a-z)=0$. The discriminant of this quadratic equation $a^{2}(4 a+9-4 z)$ is a perfect square, so $(4 a+9-4 z)=(2 t+1)^{2}$. Thus, $a=t^{2}+t+z-2$ and from the equation for $b$ we obtain that $b=a(t-1)$. Since $t \geq 2$ we have $(x-y)^{2}=a^{2}-4 a(t-1)<(a-2(t-1))^{2}$. From the other hand $a \geq t^{2}$ and therefore $a^{2}-4 a(t-1) \geq(a-2(t-1)-2)^{2}$. Since $a^{2}-4 a(t-1) \neq(a-2(t-1)-1)^{2}$ (the two numbers are of different parity) it follows that $(x-y)^{2}=a^{2}-4 a(t-1)=(a-2(t-1)-2)^{2}$.

Thus, $a=t^{2}$ i.e. $t+z=2$ and so $t=2, z=0$, which implies $a=b=4$. Therefore $x=y=2$ and these are the only positive integers satisfying the condition of the problem.
