... that no three consecutive terms are the same and amn = aman for all m, n. (Australian 1999) 173. The sequence a1, a2, a3, has a1 = 0 and an+1 = ± (an + 1) for all n. Show that ... 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 3 is at least 1/4.(USAMO ... 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 =...