91Ó°ÊÓ

a) If 11 integers are selected from \(\\{1,2,3, \ldots, 100\\}\), prove that there are at least two, say \(x\) and \(y\), such that \(0<|\sqrt{x}-\sqrt{y}|<1\) b) Write a statement that generalizes the result of part (a).

Find all real numbers \(x\) such that a) \(7\lfloor x\rfloor=\lfloor 7 x\rfloor\) b) \(\lfloor 7 x\rfloor=7\) c) \(\lfloor x+7\rfloor=x+7\) d) \(\lfloor x+7\rfloor=\lfloor x\rfloor+7\)

A rumor is spread as follows. The originator calls two people. Each of these people phones three friends, cach of whom in turn calls five associates. If no one receives more than one call, and no one calls the originator, how many people now know the rumor? How many phone calls were made?

Suppose we have seven different colored balls and four containers numbered I, II, III, and IV. (a) In how many ways can we distribute the balls so that no container is left empty? (b) In this collection of seven colored balls, one of them is blue. In how many ways can we distribute the balls so that no container is empty and the blue ball is in container II? (c) If we remove the numbers from the containers so that we can no longer distinguish them, in how many ways can we distribute the seven colored balls among the four identical containers, with some container(s) possibly empty?

. Let \(S\) be a set of seven positive integers the maximum of which is at most 24. Prove that the sums of the elements in all the nonempty subsets of \(S\) cannot be distinct.

. If \(A=\\{1,2,3,4,5,6,7\\}, B=\\{2,4,6,8,10,12\\}\), and \(f: A \rightarrow B\) where \(f=\\{(1,2),(2,6),(3,6),(4,8),(5,6)\) \((6,8),(7,12)\\}\), determine the preimage of \(B_{1}\) under \(f\) in each of the following cases. a) \(B_{1}=\\{2\\}\) b) \(B_{1}=\\{6\\}\) c) \(B_{1}=\\{6,8\\}\) d) \(B_{1}=\\{6,8,10\\}\) e) \(B_{1}=\\{6,8,10,12\\}\) f) \(B_{1}=\\{10,12\\}\)

. Let \(S \subset \mathbf{Z}^{+}\)with \(|S|=7\). For \(\emptyset \neq A \subseteq S\), let \(s_{A}\) denote the sum of the elements in \(A\). If \(m\) is the maximum element in \(S\), find the possible values of \(m\) so that there will exist distinct subsets \(B, C\) of \(S\) with \(s_{B}=s_{C}\).

Let \(k \in Z^{\top}\), Prove that there exists a positive integer \(n\) such that \(k \mid n\) and the only digits in \(n\) are 0 's and \(3^{\prime}\). .

The 50 members of Nardine's aerobics class line up to get their equipment. Assuming that no two of these people have the same height, show that eight of them (as the line is equipped from first to last) have successive heights that either decrease, or increase,

How many times must we roll a single die in order to get the same score (a) at least twice? (b) at least three times? (c) at least \(n\) times, for \(n \geq 4 ?\)