525-ps1 - Discrete Mathematics Problem Sheet 1 Pigeon-hole...

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Unformatted text preview: Discrete Mathematics Problem Sheet 1 Pigeon-hole Principle: 1. Use the pigeon-hole principle to show that one of any n consecutive inte- gers is divisible by n . 2. Use the pigeon-hole principle to show that the decimal expansion of a rational number must, after some point, become periodic. 3. The circumference of two concentric disks is divided into 200 sections each. For the outer disk, 100 of the sections are painted red and 100 of the sections are painted white. For the inner disk the sections are painted red or white in an arbitrary manner. Show that it is possible to align the two disks so that 100 or more of the sections on the inner disk have their colors matched with the corresponding sections on the outer disk. 4. Given 20 French, 30 Spanish, 25 German, 20 Italian, 50 Russian and 17 English books, how many books must be chosen to guarantee that at least (a) 10 books of one language were chosen? (b) 6 French, 11 Spanish, 7 German, 4 Italian, 20 Russian, or 8 English were chosen? 5. If there are 104 different pairs of people who know each other at a party of 30 people, then show that some person has 6 or fewer acquaintances....
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525-ps1 - Discrete Mathematics Problem Sheet 1 Pigeon-hole...

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