When will $\min_{x} \sum_{k=1}^n a_k \cos(k x) \leq 0$, given a set of positive integers $a_k$?

Question

$n$ is a fixed positive integer. Consider a set of non-negative integers $\{a_k\}_{k=1}^n$ which sums up to some constant $a > 0$, i.e., $\sum_{k=1}^n a_k = a$. I wonder if there is alway an $x$ such that $$\min_{x} \sum_{k=1}^n a_k \cos(k x) \leq 0.$$ If not, what would be the worst $\{a_k\}$ such that $$\min_{x} \sum_{k=1}^n a_k \cos(k x)$$ is maximized?

Answer 1

$f(x) = \sum_{k=1}^n a_k \cos(k x)$ is continuous and satisfies $\int_0^{2\pi} f(x) \, dx = 0$. It follows that $f(x) \le 0$ for some $x \in [0, 2\pi]$, i.e. the minimum is always $\le 0$. (Actually it is strictly negative, unless the function is identically zero.)

This holds for arbitrary coefficients $a_k$, not only if they are non-negative integers.