# The Octopus Polynomial

Algebra Level 4

Suppose $f(x)$ is a degree-$8$ polynomial such that $f(2^i)=\frac{1}{2^i}$ for all integers $0 \leq i \leq 8$. If $f(0)= \frac{a}{b}$, where $a$ and $b$ are coprime positive integers, what is the value of $a+b?$

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