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# Pigeonhole Principle

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, 9 months ago

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Is the upper bound supposed to be $$\dfrac{1}{\sqrt{3}}$$ instead of $$\sqrt{3}$$? · 3 years, 9 months ago

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... · 3 years, 9 months ago

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