TITLE: Maximum value of the lowest sum in a set of numbers
QUESTION [5 upvotes]: Last year in a maths contest held in Catalonia called Cangur it was posed the following qüestion:

We write numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10, in a certain order around a circumference. Then we sum each number with its two neighbors and we get 10 sums. Which is the maximum value that can have the lowest of these sums?
  
  a) 14      
  b) 15      
  c) 16      
  d) 17      
  e) 18

My first approach has been to add up all the triplets. By doing that, I will get a number that won't change independently from the order of the numbers around the circumference.
$$
(1+2+3)+(2+3+4)+(3+4+5)+\dots+(10+1+2)=
\sum\limits_{n=1}^{10} 3n=165
$$
Therefore, I can deduce that the average sum of every triplet is $16.5$ which automatically discards $17$ and $18$ from being the correct answer. That's because if one of the triplets added up $17$, in order to keep the average at $16.5$, we would need some other triplet to add up less than $17$, which would imply that this triplet adding up $17$ wasn't the lowest triplet. And the same explanation would work for $18$.
I can also discard $16$ because in order to keep the average at $16.5$ we would need more or half of the triplets to add up $16$. And, although I'm not 100% sure about this assertion, I think it can be proven that there aren't 5 possible ways of adding up $16$ that can be written around the circumference.
In any case, even if my last guess was true, I'm not sure how to proceed at this point. I don't know how to choose between $14$ and $15$ and I don't like to think that the only way of finding the correct answer is to try out and see if it works. So if some of you have found any other ways of facing the problem I'd really appreciate it if you shared it. Thanks!
Edit: In order to understand better what I meant by these 10 sums of triplets, here's a sketch that shows that visually for the first three sums. I would also want to make clear that the order of the numbers in the drawing isn't the order in which they should be to solve the problem.

REPLY [2 votes]: The lower sum cannot be greater that $16.5$. So the lower sum can be less or equal than 16.
If it was 16 then we must have 5 sums equal to 16 and the other 5 equal to 17 something that cannot be achieved.
Maximum lower sum equal to 15 can be achieved: 1 6 8 3 7 5 9 2 4 10 around the circle
Hence the answer is 15.