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Hong Kong competition problems

Algebra Difficulty 4.7 AIME Prove it Hong Kong

Let $\theta_1, \theta_2, \dots, \theta_{2008}$ be real numbers. Find the maximum value of
$\sin \theta_1 \cos \theta_2 + \sin \theta_2 \cos \theta_3 + \dots + \sin \theta_{2007} \cos \theta_{2008} + \sin \theta_{2008} \cos \theta_1.$

Solution

The maximum value is $1004$ .
Note that $\sin \theta_j \cos \theta_{j+1} \le \frac{1}{2}(\sin^2 \theta_j + \cos^2 \theta_{j+1})$ for any $j$ , where the indices are taken modulo $2008$ . It follows that
$\sum_{j=1}^{2008} \sin \theta_j \cos \theta_{j+1} \le \frac{1}{2} \sum_{j=1}^{2008} (\sin^2 \theta_j + \cos^2 \theta_j) = \frac{1}{2} \sum_{j=1}^{2008} 1 = 1004.$
Equality holds when $\theta_j = \frac{\pi}{4}$ for each $j$ .

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