HongKong 2022-23 IMO Selection Tests

Answer: 89

We have $${\left(\frac{x+y}{x-y}\right)}^2=\frac{x^2+y^2+2xy}{x^2+y^2-2xy}=\frac{\left(\frac{3961}{1980}+2\right)xy}{\left(\frac{3961}{1980}-2\right)xy}=7921=89^2$$ and so the answer is 89. (Note that $\frac{x+y}{x-y}>0$ as $x>y>0$.)