blizzak

This question was given to me in an assignment, and I have yet to figure it out:

Show that if the first 10 positive integers are placed around a circle, in any order, there exists three integers in consecutive locations around the circle that have a sum greater than or equal to 17.

the question instructs us to use a previous question to help us solve it, which is basically to prove that one number in a range of numbers is greater than the average of the range. That was easy, it was a contradiction proof:

let a1 + a2 ... + aN be the range of numbers, and A be the average, and assume that the each number is less than the average, so a1 + a2 ... + aN < A X n

divide each side by n:

(a1 + a2 ... + aN) / n < A
but (a1 + a2 ... + aN) / n = A (the average)
therefore it is a contradiction.

my first instinct for the problem was to try and prove the problem by contradiction, that is:
let a1 - a10 be the range; they represent the first 10 integers (in any order
then:
a1 + a2 + a3 < 17
a2 + a3 + a4 < 17
a3 + a4 + a5 < 17

and so on, so each number is used 3 times around the circle
combine the equations and you get:

3(a1 + a2 ... + a10) < 17(10)
but that works out to 165 < 170, which is fine.
Anyone have any idea how to solve this problem?

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