Find all positive real numbers \((a,b,c) \leq 1\) which satisfy \[ \huge \min \Bigg\{ \sqrt{\frac{ab+1}{abc}}\, \sqrt{\frac{bc+1}{abc}}, \sqrt{\frac{ac+1}{abc}} \Bigg \} = \sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\]
Problem
Source: Philippine Mathematical Olympiad 2017
Tags: inequalities, function
31.12.2017 15:07
By symmetry, WLOG $a\ge b\ge c$, by Cauchy-Bunyakovsky-Schwarz inequality. Consider \begin{align*}\sqrt{\frac{1-a}{a}}+\sqrt{\frac{1-b}{b}}+\sqrt{\frac{1-c}{c}}&=\frac{\sqrt{(1-a)bc}+\sqrt{(1-b)ac}+\sqrt{(1-c)ab}}{\sqrt{abc}} \\&= \frac{\sqrt{(1-a)bc}+\sqrt{a}(\sqrt{(1-b)c}+\sqrt{b(1-c)})}{\sqrt{abc}} \\&\le\frac{\sqrt{(1-a)bc}+\sqrt{a}\sqrt{(1-b)+b}\sqrt{(1-c)+c}}{\sqrt{abc}} \\&= \frac{\sqrt{(1-a)bc}+\sqrt{a}}{\sqrt{abc}}\\&\le\frac{\sqrt{(1-a)+a}\sqrt{bc+1}}{\sqrt{abc}}\\&=\sqrt{\frac{bc+1}{abc}}\\&=\huge \min \Bigg\{ \sqrt{\frac{ab+1}{abc}}, \sqrt{\frac{bc+1}{abc}}, \sqrt{\frac{ac+1}{abc}} \Bigg \}\end{align*}So inequality holds when $\frac{1-b}{b}=\frac{c}{1-c}$ and $\frac{1-a}{a}=bc$ We get, $(a,b,c)=(\frac{1}{-t^2+t+1},t,1-t)$ for $\frac{1}{2}\le t<1$ and its permutation.
01.01.2018 15:20
19.06.2018 13:28
If I were them, I would stay quiet, to let such situation happen again. There are so many math contests, that repeating some task is obvious. I think this triple looks better $\left(\frac{t^2}{t^2+t-1},1-\frac{1}{t},\frac{1}{t}\right), t>1$ How did you get this $t\ge 0.5$, which in my version would be $t\le 2$?
20.06.2018 02:03
WolfusA wrote: If I were them, I would stay quiet, to let such situation happen again. There are so many math contests, that repeating some task is obvious. Well, we actually plan to propose problems for the olympiad next year so it doesn’t happen.
21.06.2018 22:40
By "we" you mean a committee or ex-contestants?
22.06.2018 12:26
WolfusA wrote: By "we" you mean a committee or ex-contestants? We’re a bunch of ex-contestants compiling a shortlist of problems.