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 \(-9+3\sqrt{13}\)