# Law of Cosines

**Geometry**Level 5

Suppose that the angles of \(\triangle ABC\) satisfy \(\cos(3A)+\cos(3B)+\cos(3C)=1.\) Two sides of the triangle have lengths 10 and 13. There is a positive integer \(m\) so that the maximum possible length for the remaining side of \(\triangle ABC\) is \(\sqrt{m}.\) Find \(m.\)