ITSU 2011: Computational Mathematics Page 1 of 2

ITSU2011

Computational Mathematics

Activity 09

Victorian Institute of Technology Pty Ltd

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ITSU2011 – Computational Mathematics Page 2 of 2

Part A)

1) If Mike has an infinite number of red, blue, yellow, and black socks in a drawer, how many socks must the Martian pull out of the drawer to guarantee he has a pair?

2) Suppose ?? is a set of ?? + 1 integers. Prove that there exist distinct ??,?? ? ?? such that ?? ? ?? is a multiple of ??.

3) Show that in any group of ?? people, there are two who have an identical number of friends within the group.

4) Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of 5?

a. ½ b. 3/5 c. 2/3 d. 4/5 e. 1

5) Seven-line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle.

6) Prove that having 100 whole numbers, one can choose 15 of them so that the difference of any two is divisible by 7.

Part B)

1) Expand

8

2 2

3 ? ?

? ? ?

? ? x

x .

2) Find the coefficient of 3

x in the expansion of

5

3

2

1 4 ?

?

? ? ?

? +

x x .

3) What is the constant term in the expansion of

12

3

5 2 ?

?

? ? ?

? ?

x x ?

Applied Sciences