Waste less time on Facebook — follow Brilliant.

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
4 years, 2 months ago

No vote yet
2 votes


Sort by:

Top Newest

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

Jimmy Kariznov - 4 years, 2 months ago

Log in to reply

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 - 4 years, 2 months ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...