Combinatorics (Fall 2010)/Problem set 2: Difference between revisions
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== Problem 1 == | == Problem 1 == | ||
# 对任意的整数序列 <math>a_1,a_2,\ldots, a_n</math>， 证明存在 <math>1\le j\le k\le n</math> 使得 <math>\sum_{i=j}^k a_i</math> 被 <math>n</math> 整除。 | |||
# ( Erdős' favorite ) 令 <math>A\subset\{1,2,\ldots,2n\}</math> 且 <math>|A|=n+1\,</math>。则总存在不相等的 <math>j,k\in A</math> 有 <math>k\,</math> 被 <math>j\,</math> 整除。 | |||
提示：用鸽笼原理。 | |||
== Problem 2 == | == Problem 2 == | ||
Revision as of 09:39, 14 October 2010
Problem 1
- 对任意的整数序列 [math]\displaystyle{ a_1,a_2,\ldots, a_n }[/math]， 证明存在 [math]\displaystyle{ 1\le j\le k\le n }[/math] 使得 [math]\displaystyle{ \sum_{i=j}^k a_i }[/math] 被 [math]\displaystyle{ n }[/math] 整除。
- ( Erdős' favorite ) 令 [math]\displaystyle{ A\subset\{1,2,\ldots,2n\} }[/math] 且 [math]\displaystyle{ |A|=n+1\, }[/math]。则总存在不相等的 [math]\displaystyle{ j,k\in A }[/math] 有 [math]\displaystyle{ k\, }[/math] 被 [math]\displaystyle{ j\, }[/math] 整除。
提示：用鸽笼原理。