# INEQUALITY'S BLOG

## November 30, 2010

### van khea

Filed under: the new math — KKKVVV @ 10:26 pm

Let $a, b, c, r, s$ be positive real numbers such that $a\leq b\leq c$ or $a\geq b\geq c$. Prove that:

$a^{r+s}+b^{r+s}+c^{r+s}\geq a^rb^s+b^rc^s+c^ra^s$

## November 26, 2010

### Problem 72: van khea

Filed under: the new math — KKKVVV @ 8:52 am

Let $a, b, c$ be positive real numbers such that $a\geq b\geq c$ and $abc=1$. Prove that:
$\displaystyle \frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\geq \frac{3}{\sqrt{2}}$

### Problem 71: van khea

Filed under: the new math — KKKVVV @ 8:50 am

Let $a, b, c$ be positive real numbers such that $a\leq b\leq c$. Prove that:
$\displaystyle \frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\leq \sqrt{\frac{3}{2}(a+b+c)}$

## November 22, 2010

### Problem 70: van khea

Filed under: the new math — KKKVVV @ 2:40 pm

Prove that for all $a, b, c\geq 0$ we have:
$a^4+b^4+c^4+(a^2+b^2+c^2)^2\geq 4(a^3b+b^3c+c^3a)$

### Problem 69: van khea

Filed under: Uncategorized — KKKVVV @ 2:37 pm

Let $a, b, c>0$. Prove that:
$3(a^4+b^4+c^4)+a^2b^2+b^2c^2+c^2a^2\geq 4(a^3b+bc^3+c^3a)$

### Problem 68: van khea

Filed under: Uncategorized — KKKVVV @ 1:10 am

Let $a, b, c>0; abc=8$. Prove that:
$\displaystyle \frac{a^3\sqrt{3a+4b+3c}}{4a+3b+4c}+\frac{b^3\sqrt{3a+3b+4c}}{4a+4b+3c}+\frac{c^3\sqrt{4a+3b+3c}}{3a+4b+4c}\geq \frac{24\sqrt{5}}{11}$

### Problem 67: van khea

Filed under: Uncategorized — KKKVVV @ 1:08 am

Let $a, b, c>0 ; abc=8$. Prove that
$\displaystyle \frac{\sqrt[3]{7a+5b}}{c^4(2b+3c)}+\frac{\sqrt[3]{7b+5c}}{a^4(2c+3a)}+\frac{\sqrt[3]{7c+5a}}{b^4(2a+3b)}\geq \frac{3\sqrt[3]{3}}{40}$

### Problem 66: van khea

Filed under: Uncategorized — KKKVVV @ 1:07 am

Let $a, b, c>0 ; abc=8$. Prove that:
$\displaystyle \frac{\sqrt[3]{a+2b}}{c^3\sqrt{2a+b}}+\frac{\sqrt[3]{b+2c}}{a^3\sqrt{2b+c}}+\frac{\sqrt[3]{c+2a}}{b^3\sqrt{2c+a}}\geq \frac{3}{8\sqrt[6]{6}}$

### Problem 65: van khea

Filed under: Uncategorized — KKKVVV @ 1:05 am

Let $a, b, c$ be positive real numbers and satisfying $abc=8$. Prove that:
$\displaystyle \frac{a^{10}\sqrt{a+b}}{b^2+c^2}+\frac{b^{10}\sqrt{b+c}}{c^2+a^2}+\frac{c^{10}\sqrt{c+a}}{a^2+b^2}\geq 3.2^8$

### Problem 64: van khea

Filed under: Uncategorized — KKKVVV @ 1:03 am

8) Let $a, b, c$ be positive real numbers and satisfying $abc=8$. Prove that:
$\displaystyle \frac{a^3\sqrt{b+c}}{a+b}+\frac{b^3\sqrt{c+a}}{b+c}+\frac{c^3\sqrt{a+b}}{c+a}\geq 12$

### Problem 63: van khea

Filed under: Uncategorized — KKKVVV @ 1:00 am

7) Let $a, b, c>0$ ដែល $abc=1$. Prove that:
$\displaystyle P=\frac{a^4\sqrt[3]{5a+7b}}{\sqrt{a+b}}+\frac{b^4\sqrt[3]{5b+7c}}{\sqrt{b+c}}+\frac{c^4\sqrt[3]{5c+7a}}{\sqrt{c+a}}\geq 3\sqrt[3]{3}.\sqrt[6]{2}$

### Problem 62: van khea

Filed under: Uncategorized — KKKVVV @ 12:48 am

Let $a, b, c>0 ; abc=1$ . Prove that
$\displaystyle \frac{a^3\sqrt{2a+b}}{(a+b)^2}+\frac{b^3\sqrt{2b+c}}{(b+c)^2}+\frac{c^3\sqrt{2c+a}}{(c+a)^2}\geq \frac{3\sqrt{3}}{4}$

## November 21, 2010

### Problem 62: van khea

Filed under: Uncategorized — KKKVVV @ 6:29 am

6) Let $a, b, c$ be positive real numbers satisfying $abc=8$ ។ Find Min(P):
$\displaystyle P=\frac{a^5\sqrt{2a+b}}{a+b}+\frac{b^5\sqrt{2b+c}}{b+c}+\frac{c^5\sqrt{2c+a}}{c+a}$

### Problem 61: van khea

Filed under: Uncategorized — KKKVVV @ 6:20 am

5) Let $a, b, c>0 ; a^3+b^3+c^3=3 ; x, y, z\in (0, 1) ; x^3+y^3+z^3=2$ ។ Prove that:
$\displaystyle \frac{a^3}{\sqrt{(1+ab^2)(1+xy^2)}}+\frac{b^3}{\sqrt{(1+bc^2)(1+yz^2)}}+\frac{c^3}{\sqrt{(1+ca^2)(1+zx^2)}}\geq \frac{9\sqrt{2}}{10}\sqrt{1+xyz}$

### Problem 60: van khea

Filed under: Uncategorized — KKKVVV @ 6:16 am

Let $a, b, c>0 ; x, y, z\in [0, 1]$ .Prove that:
$\displaystyle a^2\sqrt{2+x+y}+b^2\sqrt{2+y+z}+c^2\sqrt{2+z+x}\geq \frac{\sqrt{2}}{3}(a+b+c)^2\sqrt{1+\sqrt[3]{xyz}}$

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