3rd Eötvös Competition Problems 1896

1.  For a positive integer n, let p(n) be the number of prime factors of n. Show that ln n ≥ p(n) ln 2. 2.  Show that if (a, b) satisfies a2 - 3ab + 2b2 + a - b = a2 - 2ab + b2 - 5a + 7b = 0, then it also satisfies ab - 12a + 15b = 0. 3.  Given three points P, Q, R in the plane, find points A, B, C such that P is the foot of the perpendicular from A to BC,