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Trigonometric equation and y=0

Source:

10/3/2010
(HUN4)(HUN 4)IMO2 If a1,a2,...,ana_1, a_2, . . . , a_n are real constants, and if y=cos⁡(a1+x)+2cos⁡(a2+x)+⋯+ncos⁡(an+x)y = \cos(a_1 + x) +2\cos(a_2+x)+ \cdots+ n \cos(a_n + x) has two zeros x1x_1 and x2x_2 whose difference is not a multiple of π\pi, prove that y=0.y = 0.
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