Contest Math: Finding maximum value under restrictions

Using Holder is straightforward, $$1 \geqslant (p^5+q^5)^{2/5} \cdot (r^5+s^5)^{3/5} \geqslant (p^2r^3+q^2s^3)$$

Equality is possible when $p=q=r=s=\frac1{\sqrt[5]2}$, so that's the maximum. Any details you need, you should ask for.