18th Eötvös Competition Problems 1911

1.  Real numbers a, b, c, A, B, C satisfy b2 < ac and aC - 2bB + cA = 0. Show that B2 ≥ AC. 2.  L1, L2, L3, L4 are diameters of a circle C radius 1 and the angle between any two is either π/2 or π/4. If P is a point on the circle, show that the sum of the fourth powers of the distances from P to the four diameters is 3/2.