Let $ a, b, c$ be positive reals. Find the smallest value of \[ \frac {a + 3c}{a + 2b + c} + \frac {4b}{a + b + 2c} - \frac {8c}{a + b + 3c}. \]
Problem
Source: CGMO 2004 P2
Tags: inequalities, inequalities unsolved
28.12.2008 08:23
Lets consider a=b=c and then we have 4/4+4/4-8/2 which is equal to 1+1-2 and the answer is 0[/img][/list][/code]
28.12.2008 08:34
I gave them the value of 1, but it they can take any value and the result is the same
28.12.2008 08:46
By computer we can compute the minimum value, and that is incorrect... I really have no good way to approach this inequality (the min occurs at $ (a,b,c)=(3-2\sqrt{2}, \sqrt{2}-1,\sqrt{2})$, for example) I think that the easiest solution is set $ a+b+c=1$ and use lagrange multipliers (clearly that makes it such that all hypotheses of that technique are satisfied).
29.12.2008 01:18
Approach and comment by Anh Cuong: Take $ x = a + 2b + c;y = a + b + 2c;z = a + b + 3c$. Then $ A = \frac {2y}{x} + \frac {4x}{y} + \frac {4z}{y} + \frac {8y}{z} - 17 \geq 12\sqrt {2} - 17$ Hm,China MO is usually hard and interesting but it seems that it is quite heavy in computation,even for the girl
22.10.2016 16:57
\bump It came in NMTC Junior final level this year. It was copied from this contest!!
01.07.2018 16:54
April wrote: Let $ a, b, c$ be positive reals. Find the smallest value of \[ \frac {a + 3c}{a + 2b + c} + \frac {4b}{a + b + 2c} - \frac {8c}{a + b + 3c}. \] Find all triples $(a, b, c)$ of positive real numbers for which the expression $$K =\frac {a + 3c}{a + 2b + c} + \frac {4b}{a + b + 2c} - \frac {8c}{a + b + 3c}$$obtains its minimum value.
04.07.2018 22:59
Here you are $(a,b,c)=\left(\frac{5\sqrt2-7}{2-\sqrt2}c,\ \frac{\sqrt2-1}{\sqrt2}c,\ c\right)=\left(\frac{(\sqrt2-1)^2}{\sqrt2}c,\ \frac{\sqrt2-1}{\sqrt2}c,\ c\right)$ for any $c>0$
30.10.2019 07:22
arqady wrote: Approach and comment by Anh Cuong: Take $ x = a + 2b + c;y = a + b + 2c;z = a + b + 3c$. Then $ A = \frac {2y}{x} + \frac {4x}{y} + \frac {4z}{y} + \frac {8y}{z} - 17 \geq 12\sqrt {2} - 17$ Hm,China MO is usually hard and interesting but it seems that it is quite heavy in computation,even for the girl Very inteligent solution for this noncyclic inequality(generating more example,generalizations..)!
17.12.2019 02:05
29.01.2021 05:09
Determine the minimum value of $$\dfrac{3a-11b-10c}{6a+12b+8c} + \dfrac{11b+4c}{24a+4b+16c} - \dfrac{a-2c}{6a+9b+3c}$$where $a,b,c>0$. ktom 2018