<s>[INST] Help me solve this olympiad challenge? Given:
$$p(x) = x^4 - 5773x^3 - 46464x^2 - 5773x + 46$$
What is the sum of all arctan of all the roots of $p(x)$?
 [/INST] Nothing additional, but for the OP a more elementary approach.
$(x-a_1)(x-a_2)(x-a_3)(x-a_4) = x^4 - c_1x^3 + c_2x^2 - c_3x + a_1a_2a_3a_4.$ Where 
$c_1 = \Sigma a_i$, $c_2 = \Sigma_{i<j}a_ia_j$ and  $c_3 = \Sigma_{i <j < k}a_ia_ja_k$.   Let $t_i=\tan^{-1}a_i.$ You want to compute $\Sigma t_i.$ Here, the trick is knowing the $\tan(x+y)$ expansion 
$$\tan(x+y) = \frac{\tan x + \tan y}{1- \tan x \tan y}.$$
Apply twice for $x=t_1+t_2$ and $y=t_3+t_4$.
$$\tan(t_1+t_2+t_3+t_4) = \frac{\tan(t_1+t_2) + \tan(t_3+t_4)}{1+\tan(t_1+t_2)\tan(t_3+t_4)}=\frac{\frac{a_1 + a_2}{1- a_1a_2}+\frac{a_3 + a_4}{1- a_3a_4}}{1-\frac{a_1 + a_2}{1- a_1a_2}\frac{a_3 + a_4}{1- a_3a_4}} = \frac{(a_1+a_2)(1-a_3a_4)+(a_3+a_4)(1-a_1a_2)}{(1-a_1a_2)(1-a_3a_4)-(a_1+a_2)(a_3+a_4)}=\frac{c_1-c_3}{1+c_4-c_2}.$$  For this polynomial $c_1=c_3$, thefore $\tan\Sigma t_i =0,$ so is $\Sigma t_i =0$
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