Given $7$ non negative reals such that $a+b+c+d+e+f+g=1$ and $$M=\max(a+b+c,b+c+d,c+d+e,d+e+f,e+f+g)$$
What is the minimum value $M$ can take
Weird thing is, this was given in Combinatorics chapter and I can't relate any topic (Pigeon Hole, Permutation & combination and Seive Formula) to this problem. Looks more like inequality problem.
Further, a hint is given which doesnt seem very helpful
Hint: Append the four numbers $a,a+b,f+g,g$ to the five given
Immediately $M\ge\frac{1}{3}$, because $(a+b)+(c+d+e)+(f+g)=1$ the least maximum value of these is $\frac{1}{3}$. If $a=d=g=\frac{1}{3}$ then $M=\frac{1}{3}$ so the minimum value of $M$ is $\frac{1}{3}$.