RMO Practice Problems - Algebra 1

Algebra Level 5

Let $P(x)$ be a polynomial of degree 2015 which satisfies $P(k) = 2^{k}$ for all $k=0,1,2,3, \ldots, 2015$ . If the value of $P(2016)$ can be expressed as

$a^{b} - c,$

where $a$ and $c$ are coprime integers and $a$ is a prime, find the value of $a+b+c$ .

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