# mt5 - Show that no matter how we arrange the rugs there...

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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. 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 ? 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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Unformatted text preview: Show that no matter how we arrange the rugs, there will be two rugs that overlap with each other over a portion of 1 9 square meters. 4. How many natural numbers less than 10 , 000 have no divisor in the set { 6 , 8 , 20 } . 5. Count the solutions in positive integers x 1 ,...,x k of the equation x 1 + ··· + x k = n . 6. Bonus Question! Given ﬁve types of coins (penny, nickel, dime, quarter and half-dollar), in how many ways can we choose 12 coins such that no coin is chosen more than 4 times ?...
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