Pigeonhole Principle

please help me

Given 7 real numbers, show that there are two of them, call it $a$ and $b$ , that always satisfy : $0<\frac{a-b}{ab+1}<\sqrt3$

my friend tell me that we can substitute $a$ with $\tan x$ and $b$ with $\tan y$ . Then what?

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

draw the tan- graph in the region [-90,+90], now divide it in 6 region(30degrees each),replace the number with tan x,thus there would be two numbers,tan m and tan n where (WLOG) 0<m-n<30degree...

Soham Chanda - 7 years, 10 months ago

Is the upper bound supposed to be $\dfrac{1}{\sqrt{3}}$ instead of $\sqrt{3}$ ?

Jimmy Kariznov - 7 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$