MathDB

Cosine Equations

Source: 2014 AIME II Problem #12

March 27, 2014

trigonometryAMCAIMEtrig identitiestangent half angleAIME II

Problem Statement

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

\sqrt{m}

. Find

m