HMMT 2012 U19

Guts
Answer

Given that $P$ is a real polynomial of degree at most 2012 such that $P(n)=2^{n}$ for $n=1,2, \ldots, 2012$ , what choice(s) of $P(0)$ produce the minimal possible value of $P(0)^{2}+P(2013)^{2}$ ?

Dataset text NuminaMath olympiads. Quoted with source.

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1-2^{2012}

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