Easy Cosinus

If $x_1=x_2=x_3=0$ , $x_4=x_5=x_6=x_7=x_8=x_9=\tfrac23\pi$ then $\cos x_j = 1$ for $1 \le j \le 3$ and $\cos x_j = -\tfrac12$ for $4 \le j \le 9$ , so that $\sum_{j=1}^9 \cos x_j = 0$ . But $\cos 3x_j = 1$ for all $1 \le j \le 9$ , and hence $\sum_{j=1}^9 \cos 3x_j = 9$ in this case. Since it is obvious that $\sum_{j=1}^9 \cos3x_j \le 9$ for all $x_j$ , we deduce that the required maximum value is $\boxed{9}$ .