Need help to prove pigeonhole problem

Question

If we pick n+1 different positive integers with every integer is less than 2n. Prove that we can always find three numbers among these n+1 numbers that one is equal to the sum of the other two numbers.

These n+1 numbers will be chosen from 1 to 2n-1 totally 2n-1 choices. I am trying to do it by using the pigeonhole principle but I don't know how to set pigeons and holes in this case. Can someone help me? Thanks

You haven't actually said what you want to prove about these integers that you're choosing. — Brian M. Scott, Oct 17 '13 at 07:35
Maybe he is trying to solve http://math.stackexchange.com/questions/501320/puzzles-or-short-exercises-illustrating-mathematical-problem-solving-to-freshmen/501327#501327. Maybe. — azimut, Oct 17 '13 at 07:47

score 1 · Accepted Answer · answered Oct 17 '13 at 12:06

1

hint: let the largest number in the set be $k<2n$.

Apply pigeonhole principle to the pairs of numbers which add up to $k$.

answered Oct 17 '13 at 12:06

Calvin Lin

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Need help to prove pigeonhole problem

1 Answers1