SE-SKOLORNAS 1964 Problem 5

$a_1, a_2, ... , a_n$ are constants such that $f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0$ for all $x$ . We seek estimates of $a_1$ . If $n = 2$ , find the smallest and largest possible values of $a_1$ . Find corresponding estimates for other values of $n$ .