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smile01

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[交流] 【求助】求解几个组合数学的习题.

1. Show that if 100 integers are chosen from 1,2,...,200,and one of the integers chosen is less than 16,then there are two chosen numbers such that one of them is divisible by the other.
2. Use the pigeonhole principle to prove that the decimal expansion of a rational number m/n eventually is repeating. For example,
34,478/99,900=0.34512512512512512...
3. Suppose that the mn people of a marching band are standing in a rectangular formation of m rows and n columns in such a way that in each row each person is taller than the one to her or his left. Suppose that the leader rearrange the people in each column in increasing order of height from front to back. Show that the rows are still arranged in increasing order of height from left to right.
4. A collection of subsets of {1,2,...,n} has the property that each pair of subsets has at least one element in common.Porve that there are at most 2 的n-1次方 subsets in the collection.
5. Let s be a set of 6 points in the plane, with no 3 of the points collinear. Color either red or blue each of the 15 line segments determined by the points of s.Show that there are at least two triangles determined by points of s which are either red triangles or blue triangles.(Both may be red ,or both may be blue,or one may be red and the other blue)
以上习题来自"组合数学"(第四版)
谢谢.

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