(New page: Choose 13 real numbers <math>x_1,x_2,\\ldots,x_{13}\\in\\mathbbb{R}<math/> with LaTeX Code: x_i\\neq x_j if LaTeX Code: i\\neq j . For these 13 numbers there exist at least two numbers a...) |
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| − | Choose 13 real numbers <math>x_1,x_2, | + | Choose 13 real numbers <math>x_1,x_2,\ldots,x_{13}\in\mathbb{R}</math> with <math>x_i\neq x_j</math> if <math>i\neq j</math>. For these 13 numbers there exist at least two numbers amongst them such that |
| − | + | <math>0 < \frac{x_i-x_j}{1+x_ix_j} \leq 2-\sqrt{3}</math> | |
| + | |||
| + | It's a tricky one to prove, unless you are one endowed with a certain amount of intuition (exempli gratia, Uli, et alii). Simple trigonometry and the ever overlooked Pigeonhole Principle are key tools to solving it. | ||
Revision as of 11:24, 31 August 2008
Choose 13 real numbers $ x_1,x_2,\ldots,x_{13}\in\mathbb{R} $ with $ x_i\neq x_j $ if $ i\neq j $. For these 13 numbers there exist at least two numbers amongst them such that
$ 0 < \frac{x_i-x_j}{1+x_ix_j} \leq 2-\sqrt{3} $
It's a tricky one to prove, unless you are one endowed with a certain amount of intuition (exempli gratia, Uli, et alii). Simple trigonometry and the ever overlooked Pigeonhole Principle are key tools to solving it.
