5th Swedish Mathematical Society Problems 1965

1.  The feet of the altitudes in the triangle ABC are A', B', C'. Find the angles of A'B'C' in terms of the angles A, B, C. Show that the largest angle in A'B'C' is at least as big as the largest angle in ABC. When is it equal? 2.  Find all positive integers m, n such that m3 - n3 = 999. 3.  Show that for every real x ≥ ½ there is an integer n such