For any integer $a$ , let $p_a(x)$ be the polynomial $p_a(x) = x^3 - x(3a^2 - \pi^2) + 2a(a^2 + \pi^2)$ . Let $m$ (respectively $M$ ) be the smallest (respectively largest) 3-digit natural number which is a zero of a polynomial $p_a(x)$ for some integer $a$ . What is the value of $m+M$ ?

The answer is 1098.

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Since, with $a \in \mathbb{Z}$ , $\begin{aligned} p_a(x) & = x^3 - x(3a^2 - \pi^2) + 2a(a^2 + \pi^2) \; = \; (x^3 - 3a^2x + 2a^3) + \pi^2(x + 2a) \\ & = (x + 2a)(x^2 - 2ax + a^2 + \pi^2) \; = \; (x + 2a)\big((x-a)^2 + \pi^2\big) \end{aligned}$ the only integer root of $p_a(x)$ is $-2a$ , we see that the smallest and largest possible $3$ -digit natural number roots of $p_a(x)$ are the smallest and largest even $3$ -digit natural numbers, so $m=100$ and $M=998$ . This makes the answer $\boxed{1098}$ .