PUMaC 2013 Team Problem 15

[system]

Prove:

\left|\sin a_{1}\right|+\left|\sin a_{2}\right|+\left|\sin a_{3}\right|+\ldots+\left|\sin a_{n}\right|+\left|\cos \left(a_{1}+a_{2}+a_{3}+\ldots+a_{n}\right)\right| \geq 1