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1. (1989-A4) If α is an irrational number, 0 < α < 1, is there a finite game with an honest coin such that the probability of one player winning the game is α? (An honest coin is one for which the probability of heads and the probability of tails are both 12 . A game is finite if with probability 1 it must end in a finite number of moves.) 2. (2000-A6) Let f (x) be a polynomial with integer coefficients. Define a sequence a0 , a1 , . . . of integers such that a0 = 0 and an+1 = f (an ) for all n ≥ 0. Prove that if there exists a positive integer m for which am = 0 then either a1 = 0 or a2 = 0. 1