RU-ALLRUSSIAN 1967 Problem 1

In the acute- angled triangle ABC, AH is the longest altitude (H lies on BC), M is the midpoint of AC, and CD is an angle bisector (with D on AB).

(a) If $\mathrm{AH}< = \mathrm{BM}$ , prove that the angle $\mathrm{ABC}< = 60$ .

(b) If $\mathrm{AH} = \mathrm{BM} = \mathrm{CD}$ , prove that ABC is equilateral.