Since square $P Q R S$ has an area of 900, then its side length is $\sqrt{900}=30$. Thus, $P Q=P S=30$. Since $M$ and $N$ are the midpoints of $P Q$ and $P S$, respectively, then $P N=P M=\frac{1}{2}(30)=15$. Since $P Q R S$ is a square, then the angle at $P$ is $90^{\circ}$, so $\triangle P M N$ is right-angled. Therefore, the area of $\triangle P M N$ is $\frac{1}{2}(P M)(P N)=\frac{1}{2}(15)(15)=\frac{225}{2}=112.5$.