Inequalities (Thailand Math POSN 1st elimination round 2014)

Only use any the following.

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

$\frac{4}{(a+2)^{2}+4b^{2}} + \frac{4}{(b+2)^{2}+4c^{2}} + \frac{4}{(c+2)^{2}+4a^{2}} \leq 1$

2.) Find all positive real solutions $a,b,c$ such that

$\begin{cases} 10a^{3}+b^{3} = 7ab + 20bc - 11 \\ b^{3}+20c^{3} = 7bc + 30ca - 20 \\ 34c^{3}+44a^{3} = 51ca + 20ab - 50 \\ \end{cases}$

3.) Let $a,b,c$ be positive real numbers such that $abc = 1$ . Prove that

$a^{3}b+b^{3}c+c^{3}a \geq a^{2/5}b^{3/5} + b^{2/5}c^{3/5} + c^{2/5}a^{3/5}$

4.) Let $a,b,c$ be positive real numbers such that $a+b+c+abc = 4$ . Prove that

$\left(\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{c+a}}+\frac{c}{\sqrt{a+b}}\right)^{2}(ab+bc+ca) \geq \frac{1}{2}(4-abc)^{3}$

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

$\frac{a^{9}}{bc}+\frac{b^{9}}{ca}+\frac{c^{9}}{ab} + \frac{3}{abc} \geq a^{4}+b^{4}+c^{4}+3$

Check out all my notes and stuffs for more problems!

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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