TH-TMO 2017 Problem 6

In an acute triangle $\vartriangle ABC$ , $D$ is the foot of altitude from $A$ to $BC$ . Suppose that $AD = CD$ , and define $N$ as the intersection of the median $CM$ and the line $AD$ . Prove that $\vartriangle ABC$ is isosceles if and only if $CN = 2AM$ .