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  1. Prove that if we take 6 numbers from 1,2,3,... 10, amongst the numbers selected are two whose difference is 5.

  2. Prove the set $ 3Z^+ =\{3,6,9,12,15\}$ is countable.

  3. Decision tree is given as follows. Start with root. If yes, it goes to left, otherwise to right. Now, if a student gets average score at 83. What is his grade he will get?

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closed as off-topic by Cameron Buie, Rahul, The Chaz 2.0, amWhy, T. Bongers Feb 27 at 21:31

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Welcome to Math.SE! We would love to hear your thoughts on these problems, and see your attempt as to how to solve them. :) –  anorton Jul 31 '13 at 1:31

1 Answer 1

up vote 4 down vote accepted

It is little known fact that numbers have a romantic life. It turns out that $1$ and $6$ are "partners," as are $2$ and $7$, also $3$ and $8$, and $4$ and $9$, and $5$ and $10$. So we have a total of five couples. Thus if we pick six people (numbers), then they cannot all belong to different couples: at least two of the six are partners.

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