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IMC 1995 Problem 11

Source: IMC 1995

February 19, 2021
functionreal analysis

Problem Statement

a) Prove that every function of the form f(x)=a02+cos⁡(x)+∑n=2Nancos⁡(nx)f(x)=\frac{a_{0}}{2}+\cos(x)+\sum_{n=2}^{N}a_{n}\cos(nx) with ∣a0∣<1|a_{0}|<1 has positive as well as negative values in the period [0,2π)[0,2\pi). b) Prove that the function F(x)=∑n=1100cos⁡(n32x)F(x)=\sum_{n=1}^{100}\cos(n^{\frac{3}{2}}x) has at least 4040 zeroes in the interval (0,1000)(0,1000).
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