A combinatorics problem

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I found this problem in a combinatorics book. It did not have the solution nor could I find one.Please help!!
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+e, d+e+f, e+f+g,
find the minimum possible value that M can take as a,b,c,d,e,f,g vary.
Hint:append the four numbers a, a+b, f+g, and g to the given five.
 
Last edited:
Denis said:
Well, WHY do you need the solution? What grade are you in?

There's a hint given here (which I don't follow):
http://books.google.ca/books?id=Snv...negative real numbers adding up to 1.&f=false
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I am from India and preparing for regional mathematics olympiad(the first step in India's selection procedure for IMO). I am using a book "Challenge and thrill of pre college mathematics intended for students of grades 9-12, and i am in 11th standard. The question was from the combinatorics section in that book. I do need the solution as (a):i am curious, and (b): i need to practice all types of problems and know the solutions if i wish to stand anywhere in the olympiad.
Please Help
 
Solution

Add the nine numbers a, a+b, a+b+c, b+c+d, c+d+e, d+e+f, e+f+g, f+g, and g to get 3(a+b+c+d+e+f+g) = 3. One of these nine numbers must be at least 3/9 = 1/3. Since all the numbers are nonnegative, we have a+b+c >= a+b >= a, and e+f+g >= f+g >= g. This means that one of a+b+c, b+c+d, c+d+e, d+e+f, e+f+g must be at least 1/3, i.e. M is at least 1/3. In fact, there is a choice of a,b,c,d,e,f,g that achieves this: a= d = g = 1/3, b=c=e=f = 0. Therefore, the answer is 1/3.
 
mathguy1729 said:
Add the nine numbers a, a+b, a+b+c, b+c+d, c+d+e, d+e+f, e+f+g, f+g, and g to get 3(a+b+c+d+e+f+g) = 3. One of these nine numbers must be at least 3/9 = 1/3. Since all the numbers are nonnegative, we have a+b+c >= a+b >= a, and e+f+g >= f+g >= g. This means that one of a+b+c, b+c+d, c+d+e, d+e+f, e+f+g must be at least 1/3, i.e. M is at least 1/3. In fact, there is a choice of a,b,c,d,e,f,g that achieves this: a= d = g = 1/3, b=c=e=f = 0. Therefore, the answer is 1/3.
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Thanks a lot, Mathguy! I had lost all hope of ever getting a solution.
 
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