Problem 94 Suppose that a + l/a E Q. Prove that an + 1/an E Q for all integer n > 0. Problem 95 Suppose that the sequence of integers {an} satises a0 = 0, a1 = a2 = 1, an+1 3an + anl 2 = (1)n. Prove that an is a perfect square. Problem 96 24 chairs are evenly spaced around a circular table on which are name cards for 24 guests. The guests failed to notice these cards until they have sat down, and it turns out that no one is sitting in fornt of his/her own card. Prove that the table can be rotated so that at least two of these guests are simultaneously correctly seated. (A much harder questions is: Can the table be rotated so that at least 3 guests are simultaneously seated correctly?) Problem 97 Let A be any set of 51 distinct integers chosen from 1, 2, 3,..., 100. Prove that there must be two distinct integers in A such that one divides the other. Problem 98 Given a positive integer 11, show that there exists a positive integer containing only the digits 0 and l (in decimal notation), and which is divisible by n. Problem 99 Let x1, x2, ...,x20 be integers. Prove that some of them have sum divisible by 20. Problem 100 Prove that from a set of 10 distinct two-digit numbers (in base 10), it is possible to select two disjoint subsets whose members have the same sum