Mr. and Mrs. Smith went to a party attended by 15 other couples. Various handshakes took...

 

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Mr. and Mrs. Smith went to a party attended by 15 other couples. Various handshakes took place during the party. In the end, Mrs. Smith asked each person at the party how many handshakes did they have. To her surprise, each person gave a different answer. How many hand shakes did Mr. Smith have? (Here we assume that no person shakes hand with his/her spouse and of course, himself/herself.) Problem 94 Suppose that a + 1/a E Q. Prove that an + 1/an E Q for all integer n > 0. Problem 95 Suppose that the sequence of integers {an} satisfies a0 = 0, a1 = a2 = 1, an+1 - 3an + an12= (-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 n, show that there exists a positive integer containing only the digits 0 and 1 (in decimal notation), and which is divisible by n. Mr. and Mrs. Smith went to a party attended by 15 other couples. Various handshakes took place during the party. In the end, Mrs. Smith asked each person at the party how many handshakes did they have. To her surprise, each person gave a different answer. How many hand shakes did Mr. Smith have? (Here we assume that no person shakes hand with his/her spouse and of course, himself/herself.) Problem 94 Suppose that a + 1/a E Q. Prove that an + 1/an E Q for all integer n > 0. Problem 95 Suppose that the sequence of integers {an} satisfies a0 = 0, a1 = a2 = 1, an+1 - 3an + an12= (-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 n, show that there exists a positive integer containing only the digits 0 and 1 (in decimal notation), and which is divisible by n.