, 2022

Solution:
Let $P(x) = a x^{2} + b x + c$. The given equations give us:
$$\begin{aligned} & a + b + c = 20 \\ & a - b + c = 22 \end{aligned}$$
Hence $b = -1$, $a + c = 21$, and so the final equation gives us $a c^{2} = 400$. Substituting $a = 21 - c$ and solving the cubic in $c$, we get $c = -4, 5, 20$. Of these, the smallest value $c = -4$ (and hence $P(x) = 25 x^{2} - x - 4$) ends up giving the largest value of $P(10)$.