... infinitely many natural number n such that x n -1 and x n+1 -1 both divisible by m (Bungari 2000-Pro3 third round) 46 A sequence is a1, a2, a3, is defined by a1= k, a2= 5k-2 and an+2= 3an+1- 2an, n ... value is a Another member is picked at random, independently of the first Its value is b Then a third, value c Show that the probability that a + b + c is divisible by is at least 1/4.(USAMO 1979) ... greatest odd divisor of an-1 + an-2 Show that, for sufficiently large n, an is constant and find this constant (in terms of r and s).(USAMO 1993) 69 The sequence a1, a2, , a99 has a1 = a3 = a5...