# Sines and Sums and Squares

Algebra Level 5

Let $$\{a_i\}_{i=1}^{2014}$$ be a sequence of $$2014$$ angles satisfying $$0\le a_i \le \dfrac{\pi}{2}$$ for all $$i=1\to 2014$$. Let the number of ordered $$2014$$-tuplets of solutions to the equation

$\sum_{cyc}\sin^2(a_1+a_2)=\sum_{cyc}(\sin a_1+\sin a_2)(\cos a_1+\cos a_2)$

be $$N$$. What is the last $$3$$ digits of $$N$$?

$\quad$ $$\text{Details and Assumptions}$$

$$\displaystyle\sum_{cyc}$$ denotes the cyclic sum. For example,

$\sum_{cyc}a_1a_2=a_1a_2+a_2a_3+\cdots +a_{n-1}a_n+a_na_1$

×