NZ-NZMO 2025 Problem 2

Let $ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$ , $\angle ABC = 70^{\circ}$ , and $AB = 1$ . Let $M$ be the midpoint of $BC$ . Let $D$ be the point on the extension of $AM$ beyond $M$ such that $\angle CDA = 110^{\circ}$ . Find the length of $CD$ .