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Some three consecutive numbers sum to at least $32$

The integers $1, 2, \ldots, 10$ are circularly arranged in an arbitrary order. Show that there are always three successive integers in this arrangement, whose sum is at least $17.$

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marked as duplicate by The Chaz 2.0, Sasha, Raskolnikov, Asaf Karagila, Marvis May 7 '12 at 18:14

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@TheChaz: AFAIK, no amount of rep allows someone to search through or for deleted answers, or see deleted answers listed on a user's profile. Rather, a 10k-er can see deleted answers when they happen upon them, or see deleted questions if they happen to have the url even after deletion. (Though I think we can see deleted questions listed on someone's favorites list, like mine, quite ironically.) –  anon May 7 '12 at 5:01
    
@anon: Thanks for the info. I cleaned up some previous comments after finding a duplicate. I still think there is another... –  The Chaz 2.0 May 7 '12 at 5:05

2 Answers 2

We will in fact prove that there is always three successive integers in this arrangement, whose sum is at least $18$.

Consider the circular arrangement of numbers starting from $1$ as follows. $$1 , a_1, a_2, \ldots, a_9$$ where $a_1,a_2,\ldots a_9 \in \{2,3,4,\ldots,10\}$.

Note that $1 + a_1 + a_2 + \cdots a_9 = 55$.

If $(a_1 + a_2 + a_3) \leq 17$ and $(a_4 + a_5 + a_6) \leq 17$ and $(a_7 + a_8 + a_9) \leq 17$, then $1 + a_1 + a_2 + \cdots a_9 \leq 52$. Which is clearly not the case, since they add up to $55$.

Hence, at least one of $(a_1 + a_2 + a_3)$ or $(a_4 + a_5 + a_6)$ or $(a_7 + a_8 + a_9)$ is greater than $17$.

$18$ is in fact the optimal lower bound on the sum which can be seen by considering the following arrangement: $$\{1,10,6,2,8,7,3,5,9,4\}$$ arranged circularly. The sum of any three taken in a circular fashion is at most $18$.

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Call the numbers, say in clockwise order, starting at some arbitrary point, $a_1$ up to $a_{10}$. Look at the sum $$(a_1+a_2+a_3)+(a_2+a_3+a_4)+ (a_3+a_4+a_5)+\cdots + (a_9+a_{10}+a_1) +(a_{10}+a_1+a_2).$$

Every number from $1$ to $10$ appears exactly $3$ times in this big sum. So the big sum is $3(1+2+\cdots+10)=165$. If all the sums of $3$ consecutive numbers in the circular ordering were $\le 16$, the big sum could be at most $160$. But it is $165$. So at least one of the sums of $3$ consecutives in the circular ordering is $\ge 17$.

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