HK-PC 2009

Let $\alpha_1$, $\alpha_2$, $\ldots$, $\alpha_{2008}$ be real numbers. Find the maximum value of $$\sin\alpha_1\cos\alpha_2 + \sin\alpha_2\cos\alpha_3 + \cdots + \sin\alpha_{2007}\cos\alpha_{2008} + \sin\alpha_{2008}\cos\alpha_1$$

Find the total number of solutions to the following system of equations: $$\begin{cases} a^2+bc\equiv a\pmod {37}\\ b(a+d)\equiv b\pmod {37}\\ c(a+d)\equiv c\pmod{37}\\ bc+d^2\equiv d\pmod{37}\\ ad-bc\equiv 1\pmod{37}\end{cases}$$