Give a proof by contradiction to show that if the integers 1, 2, ··· , 99, 100 , are placed randomly around a circle (without repetition), then there must exist three adjacent numbers along the circle whose sum is greater than 152.
Thus, we assume the following and try to show a contradiction exists:
Assume there exists a way to arrange the numbers such that their sum is less than or equal to 152.
I've tried quite a few different approaches that didn't succeed.
Many of you hinted to the following approach (which I have tried unsuccessfully). It's probably best that you attempt it separately before this taints your view of the problem:
Let $x_i$ denote the number at the $i^{th}$ position.
$ x_1 + x_2 + x_3 \le 152 $ $ x_2 + x_3 + x_4 \le 152 $ $ etc. $ $ x_{99} + x_{100} + x_1 \le 152 $ $ x_{100} + x_1 + x_2 \le 152 $
Summing both sides:
$ 3\sum\limits_{i=1}^{100} x_i \le 100(152) $ $ 3\sum\limits_{i=1}^{100} i \le 100(152) $ $ 3(5050) \le 15200 $ $ 15 150 \le 15200 $
This is not a contradiction.
I must be missing something. Thoughts?