Putnam 2013 B2

Let $C = \bigcup_{N=1}^\infty C_N$, where $C_N$ denotes the set of those `cosine polynomials' of the form $$f(x) = 1 + \sum_{n=1}^N a_n \cos(2 \pi n x)$$ for which:

(i) $f(x) \geq 0$ for all real $x$, and

(ii) $a_n = 0$ whenever $n$ is a multiple of $3$.

Determine the maximum value of $f(0)$ as $f$ ranges through $C$, and prove that this maximum is attained.