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Function with some periodic injectivity

Source: Bosnia and Herzegovina TST 2006 day 2 problem 3

July 14, 2016
functionalgebratrigonometry

Problem Statement

Let a1a_1, a2a_2,...,ana_n be constant real numbers and xx be variable real number xx. Let f(x)=cos(a1+x)+cos(a2+x)2+cos(a3+x)22+...+cos(an+x)2n−1f(x)=cos(a_1+x)+\frac{cos(a_2+x)}{2}+\frac{cos(a_3+x)}{2^2}+...+\frac{cos(a_n+x)}{2^{n-1}}. If f(x1)=f(x2)=0f(x_1)=f(x_2)=0, prove that x1−x2=mπx_1-x_2=m\pi, where mm is integer.
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