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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?

Note by Idham Muqoddas
3 years, 2 months ago

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Is the upper bound supposed to be \(\dfrac{1}{\sqrt{3}}\) instead of \(\sqrt{3}\)? Jimmy Kariznov · 3 years, 2 months ago

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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 · 3 years, 2 months ago

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