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482
Kenneth Rosen Edition 7 Exercise 6.2 Question 29 (Page No. 406)
Show that if $n$ is an integer with $n \geq 2,$ then the Ramsey number $R(2, n)$ equals $n.\text{(Recall that Ramsey numbers were discussed after Example}\: 13\:\text{in Section}\: 6.2.)$
Show that if $n$ is an integer with $n \geq 2,$ then the Ramsey number $R(2, n)$ equals $n.\text{(Recall that Ramsey numbers were discussed after Example}\: 13\:\text{in ...
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483
Kenneth Rosen Edition 7 Exercise 6.2 Question 28 (Page No. 406)
Use question $27$ to show that among any group of $20$ people (where any two people are either friends or enemies), there are either four mutual friends or four mutual enemies.
Use question $27$ to show that among any group of $20$ people (where any two people are either friends or enemies), there are either four mutual friends or four mutual en...
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485
Kenneth Rosen Edition 7 Exercise 6.2 Question 26 (Page No. 406)
Show that in a group of five people (where any two people are either friends or enemies), there are not necessarily three mutual friends or three mutual enemies.
Show that in a group of five people (where any two people are either friends or enemies), there are not necessarily three mutual friends or three mutual enemies.
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486
Kenneth Rosen Edition 7 Exercise 6.2 Question 25 (Page No. 406)
Describe an algorithm in pseudocode for producing the largest increasing or decreasing subsequence of a sequence of distinct integers.
Describe an algorithm in pseudocode for producing the largest increasing or decreasing subsequence of a sequence of distinct integers.
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488
Kenneth Rosen Edition 7 Exercise 6.2 Question 23 (Page No. 406)
Show that whenever $25$ girls and $25$ boys are seated around a circular table there is always a person both of whose neighbors are boys.
Show that whenever $25$ girls and $25$ boys are seated around a circular table there is always a person both of whose neighbors are boys.
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2 answers
491
Kenneth Rosen Edition 7 Exercise 6.2 Question 20 (Page No. 406)
Find an increasing subsequence of maximal length and a decreasing subsequence of maximal length in the sequence $22, 5, 7, 2, 23, 10, 15, 21, 3, 17.$
Find an increasing subsequence of maximal length and a decreasing subsequence of maximal length in the sequence $22, 5, 7, 2, 23, 10, 15, 21, 3, 17.$
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495
Kenneth Rosen Edition 7 Exercise 6.2 Question 16 (Page No. 405)
How many numbers must be selected from the set $\{1, 3, 5, 7, 9, 11, 13, 15\}$ to guarantee that at least one pair of these numbers add up to $16?$
How many numbers must be selected from the set $\{1, 3, 5, 7, 9, 11, 13, 15\}$ to guarantee that at least one pair of these numbers add up to $16?$
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496
Kenneth Rosen Edition 7 Exercise 6.2 Question 15 (Page No. 405)
How many numbers must be selected from the set $\{1, 2, 3, 4, 5, 6\}$ to guarantee that at least one pair of these numbers add up to $7?$
How many numbers must be selected from the set $\{1, 2, 3, 4, 5, 6\}$ to guarantee that at least one pair of these numbers add up to $7?$
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502
Kenneth Rosen Edition 7 Exercise 6.2 Question 9 (Page No. 405)
What is the minimum number of students, each of whom comes from one of the $50$ states, who must be enrolled in a university to guarantee that there are at least $100$ who come from the same state?
What is the minimum number of students, each of whom comes from one of the $50$ states, who must be enrolled in a university to guarantee that there are at least $100$ wh...
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504
Kenneth Rosen Edition 7 Exercise 6.2 Question 7 (Page No. 405)
Let $n$ be a positive integer. Show that in any set of $n$ consecutive integers there is exactly one divisible by $n.$
Let $n$ be a positive integer. Show that in any set of $n$ consecutive integers there is exactly one divisible by $n.$
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505
Kenneth Rosen Edition 7 Exercise 6.2 Question 6 (Page No. 405)
Let $d$ be a positive integer. Show that among any group of $d + 1$ (not necessarily consecutive) integers there are two with exactly the same remainder when they are divided by $d.$
Let $d$ be a positive integer. Show that among any group of $d + 1$ (not necessarily consecutive) integers there are two with exactly the same remainder when they are div...
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506
Kenneth Rosen Edition 7 Exercise 6.2 Question 5 (Page No. 405)
Show that among any group of five (not necessarily consecutive) integers, there are two with the same remainder when divided by $4.$
Show that among any group of five (not necessarily consecutive) integers, there are two with the same remainder when divided by $4.$
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509
Kenneth Rosen Edition 7 Exercise 6.2 Question 2 (Page No. 405)
Show that if there are $30$ students in a class, then at least two have last names that begin with the same letter.
Show that if there are $30$ students in a class, then at least two have last names that begin with the same letter.
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510
Kenneth Rosen Edition 7 Exercise 6.2 Question 1 (Page No. 405)
Show that in any set of six classes, each meeting regularly once a week on a particular day of the week, there must be two that meet on the same day, assuming that no classes are held on weekends.
Show that in any set of six classes, each meeting regularly once a week on a particular day of the week, there must be two that meet on the same day, assuming that no cla...
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