Austrian Mathematical Olympiad

We have $$\frac{x+1}{y}+\frac{y+1}{x}=\frac{x+x+y}{y}+\frac{y+x+y}{x}=2\left(\frac{x}{y}+\frac{y}{x}\right)+2.$$ For $x,y>0$, the AM-GM inequality gives $$\frac{\frac{x}{y}+\frac{y}{x}}{2}\ge \sqrt{\frac{x}{y}\cdot \frac{y}{x}}=1,$$ which immediately implies the desired inequality.

Equality holds for $\frac{x}{y}=\frac{y}{x}$, i.e. $x=y=\frac{1}{2}$.