# INEQUALITY'S BLOG

## August 19, 2011

### Problem 298 Van Khea

Filed under: Problem by Van Khea — KKKVVV @ 3:12 am

Let $a, b, c, d$ be positive real numbers such that $a+b+c+d\leq 4$.Prove that

$\displaystyle \frac{1}{a^4+b^4+c^4+d^4}+\frac{1}{abcd}\geq \frac{5}{4}$

## August 14, 2011

### Problem 286 Van Khea

Filed under: Problem by Van Khea — KKKVVV @ 7:26 am

Let $a, b, c$ be positive real numbers such that $abc\leq 1$. Prove that

$\displaystyle \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a^3+b^3+c^3}\geq \frac{10}{3}$

### Problem 287 Van Khea

Filed under: Problem by Van Khea — KKKVVV @ 7:23 am

Let $a, b, c$ be positive real numbers such that $a+b+c\leq 1$. Prove that

$\displaystyle \frac{1}{a^3+b^3+c^3}+\frac{1}{abc}\geq 36$

## August 13, 2011

### Problem 272 Van Khea

Filed under: Problem by Van Khea — KKKVVV @ 5:05 pm

Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2=3$. Prove that:

$(ab)^3+(bc)^3+(ca)^3+a^2b^2c^2\leq 4$

### Problem 267 Van Khea

Filed under: Problem by Van Khea — KKKVVV @ 5:02 pm

Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2=3$. Prove that

$\displaystyle \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{2}{3}(a+b+c)+1$

### Problem 281 Van Khea

Filed under: Problem by Van Khea — KKKVVV @ 10:33 am

Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2=1$. Prove that:

$\displaystyle (ab+bc+ca)^2\leq \frac{3\sqrt{3}}{4}abc+\frac{3}{4}$

### Problem 279 Van Khea

Filed under: Problem by Van Khea — KKKVVV @ 10:20 am

Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2=3$. Prove that:

$\displaystyle ab^2+bc^2+ca^2\leq \frac{3}{4}(a+b+c+1)$

### Problem 276 Van Khea

Filed under: Problem by Van Khea — KKKVVV @ 10:02 am

Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2=1$. Prove that

$\displaystyle a+b+c-4abc\geq \frac{5}{3\sqrt{3}}$

### Problem 275 Van Khea

Filed under: Problem by Van Khea — KKKVVV @ 9:59 am

Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2=1$. Prove that

$\displaystyle \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq 6\sqrt[3]{abc}+\frac{1}{3abc}$

### Problem 274 Van Khea

Filed under: Problem by Van Khea — KKKVVV @ 9:56 am

Let $a, b, c>0\& a+b+c=3$. Prove that:

$a^2\sqrt{1+bc}+b^2\sqrt{1+ca}+c^2\sqrt{1+ab}\geq 3\sqrt{2}$

### Problem 240 Van Khea

Filed under: Problem by Van Khea — KKKVVV @ 9:50 am

Let $a, b, c$ be the lengths of the sides of a trangle. Prove that:

$\displaystyle \frac{1}{(a^2+b^2)^2}+\frac{1}{(b^2+c^2)^2}+\frac{1}{(c^2+a^2)^2}\geq \frac{9}{4(a^3b+b^3c+c^3a)}$

### Problem 285 Van Khea

Filed under: Problem by Van Khea — KKKVVV @ 9:45 am

Let $a, b, c$ be positive real numbers such that $abc\leq 1$. Prove that:

$\displaystyle \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b+c}\geq \frac{10}{3}$

## August 7, 2011

### Problem 246 van khea

Filed under: Problem by Van Khea — KKKVVV @ 1:31 am

Let $a, b, c$ be positive real numbers. Prove that:

$\displaystyle \sum_{cyc}\frac{(a-b)^2}{a\sqrt{bc}}\geq 2\sum_{cyc}\frac{a-b}{a+b}\frac{a-c}{b}+2\sum_{cyc}\frac{a-c}{a+c}\frac{a-b}{c}$

### Problem 247 van khea

Filed under: Problem by Van Khea — KKKVVV @ 1:26 am

Let $a, b, c$ be positive real numbers such that $a\leq b\leq c\&ab+bc+ca=1$. Prove that

$\displaystyle \frac{(ab)^{\frac{2}{3}}}{\sqrt{1+b^2}}+\frac{(bc)^{\frac{2}{3}}}{\sqrt{1+c^2}}+\frac{(ca)^{\frac{2}{3}}}{\sqrt{1+a^2}}\leq \sqrt[3]{\frac{9}{8}(a+b+c)}$

### Problem 251 van khea

Filed under: Problem by Van Khea — KKKVVV @ 1:21 am

Let $a, b, c$ be positive real numbers such that $a\leq b\leq c\&a+b+c=3$. Prove that:

$\displaystyle \sqrt[4]{a(a+b)}+\sqrt[4]{b(b+c)}+\sqrt[4]{c(c+a)}\geq \frac{\sqrt[4]{2}}{3}(ab+bc+ca)^2$

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