Inequality

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#1 h Aug 20, 2016, 5:14 PM • 2 Y

Y by Adventure10, Mango247

Prove(using only AM-GM if possible) that for a,b,c>0
$\sum_{cyc}\frac{c}{a+b}\leq\frac{3}{2}\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-2\right)$

This post has been edited 2 times. Last edited by kraken296, Aug 20, 2016, 5:52 PM

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arqady

#2 h Aug 20, 2016, 7:03 PM • 1 Y

Y by Adventure10

kraken296 wrote:

Prove(using only AM-GM if possible) that for a,b,c>0
$\sum_{cyc}\frac{c}{a+b}\leq\frac{3}{2}\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-2\right)$

It's $\sum_{cyc}(3a^4c^2+3a^3b^3+a^4bc-2a^3b^2c-2a^3c^2b-3a^2b^2c^2)\geq0$ , which is obvious by AM-GM.

This post has been edited 1 time. Last edited by arqady, Aug 20, 2016, 8:40 PM

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arqady

#3 h Aug 20, 2016, 7:24 PM • 1 Y

Y by Adventure10

The following inequality is stronger, but also easy.
Let $a$ , $b$ and $c$ be positive numbers. Prove that:
$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq\frac{3}{2}+\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}$

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lwwwww

#4 h Aug 20, 2016, 7:38 PM • 1 Y

Y by Adventure10

arqady wrote:

The following inequality is stronger, but also easy.
Let $a$ , $b$ and $c$ be positive numbers. Prove that:
$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq\frac{3}{2}+\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}$

Let $a$ , $b$ and $c$ be positive numbers. Prove that:
$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq\frac{a+c}{b+c}+\frac{a+b}{a+c}+\frac{b+c}{a+b}$

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arqady

#5 h Aug 20, 2016, 7:38 PM • 1 Y

Y by Adventure10

For the collection.
For all triangle prove that:
$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\geq\frac{8}{3}\left(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\right)$

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arqady

#6 h Aug 20, 2016, 7:40 PM • 1 Y

Y by Adventure10

lwwwww wrote:

Let $a$ , $b$ and $c$ be positive numbers. Prove that:
$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq\frac{a+c}{b+c}+\frac{a+b}{a+c}+\frac{b+c}{a+b}$

It's $\sum_{cyc}(a^4c^2+a^3b^3-a^3c^2b-a^2b^2c^2)\geq0$ , which is obvious.

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#7 h Aug 20, 2016, 7:46 PM • 3 Y

Y by arqady, Adventure10, Mango247

arqady wrote:

kraken296 wrote:

Prove(using only AM-GM if possible) that for a,b,c>0
$\sum_{cyc}\frac{c}{a+b}\leq\frac{3}{2}\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-2\right)$

It's $\sum_{cyc}(a^4c^2+3a^3b^3+a^4bc-2a^3b^2c-2a^3c^2b-3a^2b^2c^2)\geq0$ , which is obvious by AM-GM.

last inequality should be
$\sum_{cyc}(3a^4c^2+3a^3b^3+a^4bc-2a^3b^2c-2a^3c^2b-3a^2b^2c^2)\geq0$ ,

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sqing

#8 h Aug 21, 2016, 1:00 AM • 1 Y

Y by Adventure10

arqady wrote:

The following inequality is stronger, but also easy.
Let $a$ , $b$ and $c$ be positive numbers. Prove that:
$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq\frac{3}{2}+\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}$

http://www.artofproblemsolving.com/community/c6h1263173p6565534

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