# Inequalities (Thailand Math POSN 1st elimination round 2014)

Only use any the following.

• Basic inequalities
• AM-GM-HM
• Holder
• Cauchy Schwarz (Titu's lemma is acceptable)
• Weighted AM-GM

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!

Thailand Math POSN 2013

Thailand Math POSN 2014

Note by Samuraiwarm Tsunayoshi
5 years, 2 months ago

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I like the kind of questions where a suitably application of AM-GM gives the resultant inequality, since that relies on your ability to spot / identify patterns.

For example, here is how to do 3:

Apply AM-GM to $2 a^3 b + 2 b^3 c + c^3 a$.

Staff - 5 years, 2 months ago

Multiply $abc$ on the RHS. Then Muirhead.

- 5 years, 2 months ago

Right. Muirhead is just a fancy term for bunching AM-GM.

Staff - 5 years, 2 months ago

In addition, graders hate it because it involves no ingenuity at all.

- 5 years, 2 months ago

Sorry I miss typed no.4 here. It is $a+b+c+abc = 4$ instead of $1$.

- 5 years, 2 months ago

Is it necessary to use just the ineqs u stated?

- 5 years, 2 months ago

Yup.

- 5 years, 2 months ago