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Integer problems

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Suppose that integers 1,2,3,4,5,6,7,8,9,10 are arranged randomly along a circle.

1) show that For each circular arrangement, there exists at least three adjacent numbers whose sum is greater than 17

2) take n + 1 integers from {1,2,3,....., 2n}. Show there exist two integers, one divides the other completely.

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Solution Summary

There are two proofs, one regarding circular arrangement of integers and one regarding divisibility

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1. Proof: (By contradiction)
Suppose the sum of any three adjacent numbers is less than or equal to 17. We consider the numbers on the circle. Suppose these numbers are a1,a2,a3,...,a9,a10 and back to a1. Since a circle has no starting and ending point, we can assume a1=1. Then we have
a2+a3+a4<=17, a5+a6+a7<=17, ...

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