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【文章內(nèi)容簡(jiǎn)介】 le Principle) If k+1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects. Proof Suppose that none of the k boxes contains more than one object. Then the total number of objects would be at most k. This is a contradiction. Example 1. Among any 367 people, there must be at least two with the same birthday, because there are only 366 possible birthdays. Ch514 Example 2 In any group of 27 English words, there must be at least two that begin with the same letter. Example 3 How many students must be in a class to guarantee that at least two students receive the same score on the final exam ? (0~100 points) Sol: 102. (101+1) Theorem 2. (The generalized pigeon hole principle) If N objects are placed into k boxes, then there is at least one box containing at least objects. . 21 objects, 10 boxes ? there must be one box containing at least objects. 31021 =??????Ch515 Example 5 Among 100 people there are at least who were born in the same month. ( 100 objects, 12 boxes ) 912100 =??????Ch516 Example 10 During a month with 30 days a baseball team plays at least 1 game a day, but no more than 45 games. Show that there must be a period of some number of consecutive days during which the team must play exactly 14 games. 45sum ?存在一段時(shí)間的 game數(shù)和 =14 (跳過 ) Ch517 Sol: Let aj be the number of games played on or before the jth day of the month. (第 1天～第 j天的比賽數(shù)和 ) Then is an increasing sequence of distinct integers with Moreover, is also an increasing sequence of distinct integers with There are 60 positive integers between 1 and 59. Hence, such that )301( ?? j 3021 ,.. ., aaa ja j ??? 451 )45...1 .,.(30321 ?????? aaaaei 14,.. .,14,14 3021 ??? aa 591415 ??? ja )5914...14141415 .,.( 30321 ?????????? aaaaei 14,.. .,14,.. .,301301 ?? aaaa ji and ? )場(chǎng)14共 天天～第1第 .,.( 14 ijeiaa ji ??=(跳過 ) Ch518 Def. Suppose that is a sequence of numbers. A subsequence of this sequence is a sequence of the form where . sequence: 8, 11, 9, 1, 4, 6, 12, 10, 5, 7 subsequence: 8, 9, 12 (?) 9, 11, 4, 6 (?) Def. A sequence is called increasing (遞增 ) if A sequence is called decreasing (遞減 ) if A sequence is called strictly increasing (嚴(yán)格遞增 ) if A sequence is called strictly decreasing (嚴(yán)格遞減 ) if Naaa ,.. ., 21 miii aa ,.. ., 21 Niii m ????? ...1 21 ) .,.( 保持原順序ei1?? ii aa 1?? ii 1?? ii aa 1?? iiCh519 Theorem 3. Every sequence of n2+1 distinct real numbers contains a subsequence of length n+1 that is either strictly increasin
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