HMMT 2015 Algebra #7

Suppose $(a_1, \; a_2 , \; a_3 , \; a_4)$ is a 4-term sequence of real numbers satisfying the following two conditions:

• $a_3=a_2+a_1$ and $a_4=a_3+a_2$;

• there exist rean numbers $a, \; b, \;c$ such that 

$an^2+bn+c=\cos(a_n)$

for all $n \in \{1, \;2, \;3, \;4\}$.

Compute the maximum possible value of $\cos(a_1)-\cos(a_4)$ over all such sequences $(a_1, \; a_2, \; a_3,\; a_4)$.

정답 $-9+3\sqrt{13}$