On an extremal problem on the minimum of a trigonometric polynomial

The exact value of the quantity
$$ M(n)=\min\biggl\{-\min_x\sum_{k=1}^na_k\cos(kx)\colon a_1\geqslant 1,\dots ,a_n\geqslant 1\biggr\} $$
is found for any positive integer $n$. It is proved that an extremal trigonometric polynomial on which this minimum is attained is unique. Some properties of these extremal polynomials are studied.