$a,b,c,d,e,f,g$ 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.
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?