Question 25
Let $a \gt 0,$ and let $P(x)$ be a polynomial with integer coefficients such that $$
\begin{aligned} P(1) &= P(3) \\ &= P(5) = P(7) = a, \end{aligned}
$$ and $$
\begin{aligned} P(2)&=P(4)=P(6)\\ &=P(8)=-a. \end{aligned}
$$ What is the smallest possible value of $a?$
\begin{aligned} P(1) &= P(3) \\ &= P(5) = P(7) = a, \end{aligned}
$$ and $$
\begin{aligned} P(2)&=P(4)=P(6)\\ &=P(8)=-a. \end{aligned}
$$ What is the smallest possible value of $a?$