IMC 1995 Problem 11

a) Prove that every function of the form $$f(x)=\frac{a_{0}}{2}+\cos(x)+\sum_{n=2}^{N}a_{n}\cos(nx)$$ with $|a_{0}|<1$ has positive as well as negative values in the period $[0,2\pi)$ . b) Prove that the function $$F(x)=\sum_{n=1}^{100}\cos(n^{\frac{3}{2}}x)$$ has at least $40$ zeroes in the interval $(0,1000)$ .