MidtermB-SOLUTIONS.pdf - Introduction to Combinatorics MATHC44 Midterm\u2013B University of Toronto Scarborough Instructor Stefanos Aretakis Last Name

MidtermB-SOLUTIONS.pdf - Introduction to Combinatorics...

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Introduction to Combinatorics MATHC44 Midterm–B University of Toronto, Scarborough Instructor: Stefanos Aretakis February 14, 2018 Last Name: First Name: Student #: Instructions: Duration: 2 hours Do not turn this page and start the exam until you are told to do so. Please do not make use of personal notes, computers or books. No aids are allowed. Provide a solution to all of the seven problems in pages 2–8. Write your solution only on the designated space. You might request extra paper for your draft, but this will not be graded. Maximum points 100 (Problem 1 is worth 16 points and each other problem is worth 14 points). Good Luck! 1
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Problem 1. (16 points) The numbers 1 , 2 , 3 , · · · , 16 n - 1 are written on the blackboard. Consider the following rule: we select two numbers x, y on the blackboard, we delete them and replace them with three times the absolute value of their difference: 3 | x - y | . After applying 16 n - 2 times this rule, is it possible that 1 is the final number on the blackboard?
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