gouthamphilomath wrote: $(HUN 4)$IMO2 If $a_1, a_2, . . . , a_n$ are real constants, and if $y = \cos(a_1 + x) +2\cos(a_2+x)+ \cdots+ n \cos(a_n + x)$ has two zeros $x_1$ and $x_2$ whose difference is not a multiple of $\pi$, prove that $y = 0.$ This may be written $y=\alpha\cos(x)+\beta\sin(x)$ and so $y=c\cos(x+d)$ And if $c\ne 0$, all zeroes of $c\cos(x+d)$ have their difference multiple of $\pi$ And so $c=0$ and $y=0$ Q.E.D.