91Ó°ÊÓ

a) Show that if any 14 integers are selected from the set \(S=\\{1,2,3, \ldots, 25\\}\), there are at least two whose sum is 26 . b) Write a statement that generalizes the results of part (a) and Example 5.45.

Logic chips are taken from a container, tested individually, and labeled defective or good. The testing process is continued until either two defective chips are found or five chips are tested in total. Using a tree diagram, exhibit a sample space for this process.

A rumor is spread as follows. The originator calls two people. Each of these people phones three friends, each 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?

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.

Let \(A \subset\\{1,2,3, \ldots, 25\\}\) where \(|A|=9\). For any subset \(B\) of \(A\) let \(s_{B}\) denote the sum of the elements in \(B\). Prove that there are distinct subsets \(C, D\) of \(A\) such that \(|C|=|D|=5\) and \(s_{C}=s_{D}\).

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be the integer sequence defined recursively by 1) \(a_{1}=1\); and, 2) For all \(n \in \mathbf{Z}^{+}\)where \(n \geq 2, a_{n}=2 a_{[n / 2]}\). a) Determine \(a_{n}\) for all \(2 \leq n \leq 8 .\) b) Prove that \(a_{n} \leq n\) for all \(n \in \mathbf{Z}^{+}\).

Let \(S\) be a set of five positive integers the maximum of which is at most 9 . Prove that the sums of the elements in all the nonempty subsets of \(S\) cannot all be distinct.

How many positive integers between 1 and 30 (inclusive) must we select in order to guarantee that we have two integers-say, \(x, y\)-in our selection whose greatest common divisor is greater than 1 ?

During the first six weeks of his senior year in college, Brace sends out at least one resumé each day but no more than 60 resumés in total. Show that there is a period of consecutive days during which he sends out exactly 23 resumés.

Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be defined by \(f(x)=\lfloor x\rfloor\), the greatest integer in \(x\). Find \(f^{-1}(B)\) for each of the following subsets \(B\) of \(\mathbf{R}\). a) \(B=\\{0,1\\}\) b) \(B=\\{-1,0,1\\}\) c) \(B=[0,1)\) d) \(B=[0,2)\) e) \(B=[-1,2)\) f) \(B=[0,1]\) g) \(B=[-1,2]\) h) \(B=[-1,0) \cup(1,3]\)