# A Pigeonhole Principle problem

101 positive integers are placed on a circle whose sum is 300.Prove that it is possible to choose from these numbers some consecutive numbers whose sum is equal to 200. (I don't know if the word 'consecutive' is appropriate in this case ,I mean that these numbers follow each other on that circle)

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 Do you mean two consecutive numbers? – Daniel Littlewood Jan 20 at 9:20 @DanielLittlewood Surely not: use one hundred $1$s and a single $200$; then you cannot find two consecutive numbers whose sum is $200$. – B.D Jan 20 at 9:21 Adjacent is a better term, I think. The problem should be the same as the sum of adjacent numbers is 100. – lab bhattacharjee Jan 20 at 9:21 Yes, adjacent is a better word .I mean adjacent not consecutive. – user1978522 Jan 20 at 9:26 I think the circle here can be viewed as a string and the problem asks to prove that there is a substring whose sum is 200. – user1978522 Jan 20 at 9:31

Start at a certain position and form sums of subsequences of length $1, 2, \dotsc, 101$ starting at that position and going in clockwise direction. This is an increasing sequence of $101$ numbers so there are two different entries that are equal $\bmod$ $100$ (end in the same two digits). The difference between those entries is a positive multiple of $100$ and less than $300$ so either $100$ or $200$. This difference corresponds to a subsequence of numbers on the circle with sum either $100$ or $200$. If it is $200$ we're done, otherwise take the complement of that sequence.

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When you say "subsequence" in line 5, you mean consecutive right? – alancalvitti Jan 20 at 15:56
@alancalvitti Of course. – WimC Jan 20 at 16:49
That's what I thought. What's the general principle that enabled you to solve this problem? Would I find it in, eg, Bona's Walk Through Combinatorics? – alancalvitti Jan 20 at 17:18
@alancalvitti I've known the pigeon hole principle for more than twentyfive years, plenty of time to practice! – WimC Jan 20 at 18:21
I'm familiar with the principle, but curious how you applied it here. – alancalvitti Jan 25 at 14:50
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What is the integer sum of a circle?

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Nice answer, mate. – Git Gud Jan 20 at 10:06
Are there different boxes/ buttons to answer and to comment ? I´ll be more careful. – user55514 Jan 20 at 10:08
Yes. I don't think the OP will know how to answer your question anyway. The problem was probably put like that to him. – Git Gud Jan 20 at 10:10