Hard SUM

$x_i=sin(\alpha_i)$ .

$$

$sin(3\gamma)=3sin(\gamma)-4sin(\gamma)^3$

$$

Summing all $\alpha_i$ we get $\displaystyle \sum_{i=1}^n sin(3\alpha_i)=3\displaystyle \sum_{i=1}^n sin(\alpha_i)-4\displaystyle \sum_{i=1}^n sin(\alpha_i)^3=3\displaystyle \sum_{i=1}^n x_i-4\displaystyle \sum_{i=1}^n x_i^3=3\displaystyle \sum_{i=1}^n x_i$

$$

$sin(3\gamma)\leq1\Rightarrow 3\displaystyle \sum_{i=1}^n x_i\leq n$ and in this case, for $n=2016$ the greatest number is $\frac {n}{3}=\frac {2016}{3}=672$