1998 IMO Shortlist Problems/C2

n is a fixed positive integer. An odd n-admissible sequence a_1, a_2, a_3, ... satisfies the following conditions: 

1 : a_1 = 1. 2 : a_2k = a_2k-1 + 2 or a_2k-1 + n 3 : a_2k+1 = 2a_2k or n a_2k. An even n-admissible sequence satisfies 1 : a_1 = 1 2 : a_2k = 2a_2k-1 or n a_2k-1; (3) a_2k+1 = a_2k + 2 or a_2k + n. An integer m > 1 is n-attainable if it belongs to an odd n-admissible sequence or an even n-admissible sequence. Show that for n > 8 there are infinitely many positive integers which are not n-attainable. Show that all positive integers except 7 are 3-attainable

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