1st All Soviet Union Mathematical Olympiad 1967 Problems & Solutions

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 AH ≤ BM, prove that the angle ABC ≤ 60. (b)  If AH = BM = CD, prove that ABC is equilateral. 2. (a)  The digits of a natural number are rearranged and the resultant number is added to the original