Let be $a,b,c\in\mathbb{R}$ so that the sum of two of them is never equal to $1$. Prove that atleast on of $\frac{ab}{a+b-1},\frac{bc}{b+c-1},\frac{ca}{c+a-1}$ can't be in the interval $(0,1)$.

I aproached it with contradiction, but can't get sth that is not true.


1 Answer 1


Let $0<\frac{ab}{a+b-1}<1$, $0<\frac{ac}{a+c-1}<1$ and $0<\frac{bc}{b+c-1}<1$.

Hence, $0<\frac{a^2b^2c^2}{\prod\limits_{cyc}(a+b-1)}<1$.

In another hand, $0<1-\frac{ab}{a+b-1}<1$, $0<1-\frac{ac}{a+c-1}<1$ and $0<1-\frac{bc}{b+c-1}<1$, which gives

$0<\frac{(1-a)(b-1)}{a+b-1}<1$, $0<\frac{(1-c)(a-1)}{a+c-1}<1$ and $0<\frac{(1-b)(c-1)}{b+c-1}<1$, which gives

$0<\frac{-\prod\limits_{cyc}(a-1)^2}{\prod\limits_{cyc}(a+b-1)}<1$, which is a contradiction.


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