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$a,b,c,d,e,f,g$ are non negative real numbers adding up to $1$. If $M$ is the maximum of the five numbers$$a+b+c,b+c+d,c+d,\ \ d+e+f,e+f+g$$find the minimum possible value that $M$ can take as $a,b,c,d,e,f,g$ vary.

First of all please help me understand WHAT THIS PROBLEM MEANS? Any alternate statement for this would sort it out. And some hint for this problem, so that I can try it by myself first, and then I'll invite you to check whether it's correct or not.Thanks.

Hint from author:

Append the four numbers $a,a+b,f+g,g$ to the five given.

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    Should the 3rd term be $c+d+e$? – 2017-02-28
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    @angryavian My book don't says so, let me on-line PDF too. – 2017-02-28
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    The problem has no tie to Number Theory. So that's not a correct tag, and the title is also wrong with regard to that. – 2017-02-28
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    @quasi feel free to edit (please). – 2017-02-28
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    @angryavian Google books don't have that particular page, but I wrote exctly as written in book, they gave hint also please check in question (edited). – 2017-02-28
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    This appears to be a problem in linear programming and convex optimization; I don't see what it has to do with the pigeonhole principle. Was any relation to the pigeonhole principle clarified by the context of the stated problem? – 2017-02-28
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    @mathlover: What book is the problem from? – 2017-02-28
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    @DavidG.Stork they gave this under exercise Pigeon hole principle, and see the hint, need to append $4$ to $5$, so one will have same (from those $4's$), something like that. – 2017-02-28
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    @quasi Challenge and Thrill of pre collage Mathematics. see: https://books.google.co.in/books/about/Challenge_and_Thrill_of_Pre_College_Math.html?id=SnvBeodeTDcC – 2017-02-28
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    Of the list, could c + d be the maximum? – 2017-02-28
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    Based on my previous comment, it's almost certain that $c+d$ is a typo, and as angryavian suggested, was intended to be $c + d + e$. – 2017-02-28
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    Can we do better than $a=d=g=\frac 13$? – 2017-02-28
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    @Joffan and then if we made $b=c=e=f=0$ then it satisfy problem, and we have $M_{mim}=\frac{1}{3}$. But explain how you got that (previous comment's result). – 2017-02-28
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    assume $a+b+c = e+f+g<3/7$, what are the limits on $d$, depends on $b,c,e,f$ so try zero for those... – 2017-02-28

1 Answers 1

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I will answer the question as if $c+d$ were replaced with $c+d+e$.

As suggested by the hint, we append the four numbers to the list. This will not change the maximum since we are only considering nonnegative numbers.

The sum of the list of nine numbers is $3$. (Why?)

By the pigeonhole principle, at least one of these nine numbers must be $\ge 1/3$. Can you conclude from here?