# Tantalizing Trigonometric Equations

Algebra Level 5

For a parameter $$a,$$ suppose $$u$$ is the smallest positive real number such that $100\cos ^2 u -100\sin ^2 u =a+25$ and $$v$$ is the largest negative real number such that $25\sqrt{3}\sin v-75\cos v =a$ For how many integer values of $$a$$ is $$u+v$$ non-negative?

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