Mathematical Reflections 2017, Issue 4 - Problem J417

Problem:
Solve in positive real numbers the equation $$\dfrac{x^2+y^2}{1+xy}=\sqrt{2-\dfrac{1}{xy}}.$$

Proposed by Adrian Andreescu, Dallas, Texas, USA

Solution:
Let $s=x+y$ and $p=xy$. Then, the given equation can be written as $$\dfrac{s^2-2p}{1+p}=\sqrt{2-\dfrac{1}{p}}.$$
By the AM-GM Inequality, we have $s^2-2p \geq 2p$, so $$\sqrt{2-\dfrac{1}{p}} \geq \dfrac{2p}{1+p} \iff \dfrac{(p-1)^2(2p+1)}{p(p+1)^2} \leq 0.$$ Since $p \geq 0$, it follows that $p=1$, which gives $s=2$. So, $x+y=2$ and $xy=1$, which yields $(x,y)=(1,1)$.