mt5sol - Homework 5, Math 245, Fall 2010 Due Monday,...

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Unformatted text preview: Homework 5, Math 245, Fall 2010 Due Monday, November 1 in class. The solution of each exercise should be at most one page long. If you can, try to write your solutions in LaTex. Each question is worth 2 points. 1. Show that among any 20 distinct numbers from the arithmetic progression 1 , 4 ,..., 97 , 100, there are two whose sum equals 104. Proof. The numbers 1 , 4 ,..., 97 , 100 can be partitioned into the following 18 sets (or boxes): B 1 = { 4 , 100 } B 2 = { 7 , 97 } ......... B 16 = { 49 , 55 } B 17 = { 1 } B 18 = { 52 } . When choosing 20 numbers among 1 , 4 ,..., 97 , 100, we always pick at least 18 numbers from the set B 1 B 2 ...B 16 . This means there will be at least two numbers from the same set B i = { 1 + 3 i, 104- 1- 3 i } for some i, 1 i 16. These two numbers will add up to 104. 2. A box contains 6 red balls, 5 black balls, and 4 green balls. How many balls must we pick to ensure we have taken two balls of the same color ? How many balls must we choose to ensure we have taken out two balls with different colors ? Proof. Since we have 3 colors, choosing 4 balls will guarantee that we have two balls of the same color. Picking 3 balls does not guarantee at least two balls of the same color as we may choose 1 red ball, 1 black ball and 1 green ball. Since the largest number of balls of the same color is 6, choose 7 balls will guarantee that have two balls of different colors. Picking 6 balls does not guarantee two balls of different color as we could choose the 6 red balls. 3. We have 9 rugs of area 1 square meter that we place in a room of area 5 square meters....
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mt5sol - Homework 5, Math 245, Fall 2010 Due Monday,...

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