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# Inequalities (1st math Thailand POSN 2014)

Only use these following inequalities:

• A.M.-G.M.-H.M.
• Cauchy Schwarz
• Weighted A.M-G.M. with weight sum = 1

1.) Let $$a,b,c$$ be positive real numbers such that $\displaystyle \frac{\left(\displaystyle a+ \frac{1}{a}\right)\left(\displaystyle b+ \frac{1}{b}\right)\left(\displaystyle c+ \frac{1}{c}\right)}{abc} = 8$ Prove that $$abc \geq 1$$.

2.) Let $$n$$ be positive integers $$n \geq 2$$ and $$x_{1},y_{1},x_{2},y_{2},...,x_{n},y_{n}$$ be positive real numbers such that

$\sum\limits_{i = 1}^{n} x_{i} \geq \sum\limits_{i = 1}^{n} x_{i}y_{i}$

Prove that $\sum\limits_{i = 1}^{n} x_{i} \leq \sum\limits_{i = 1}^{n} \displaystyle \frac{x_{i}}{y_{i}}$

3.) Let $$a,b,c$$ be positive real numbers, prove that

$a^{3}+b^{3}+c^{3} \geq \frac{1}{4}(a+b+c)^{3} - 6abc$

4.) Let $$a,b,c,d$$ be positive real numbers and $$36a+4b+4c+3d = 25$$, prove that

$abcd(a+b+c+d)(ab^{1/2}c^{1/3}d^{1/4}) \leq \frac{1}{\sqrt{6}}(a^{4}b+b^{4}c+c^{4}d+d^{4}a)$

5.) Let $$a\geq 2, b\geq 6, c \geq 12$$, find and show the maximum value of

$\displaystyle \frac{bc\sqrt{a-2} + ca\sqrt[3]{b-6} + ab\sqrt[4]{c-12}}{abc}$

and find the conditions of $$a,b,c$$ to be maximum.

This is the part of Thailand 1st round math POSN problems.

Note by Samuraiwarm Tsunayoshi
2 years, 7 months ago

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For number 1, using the fact that a + (1/a) >= 2, this also applies for b and c. Hence, [a + (1/a)][b + (1/b)][c + (1/c)] >= 8. Hence, 8abc >= 8 and abc >= 1. · 2 years, 7 months ago

This is the easiest question because professor let everyone get at least 1 problem to bring some joy of solving the hardest topic of all. · 2 years, 7 months ago

Yep, that's how I did it. · 2 years, 7 months ago

first one is easy · 2 years, 7 months ago

Can someone tell me how to solve 4? I solved the rest. · 1 year, 7 months ago

Never mind. Just realised that 4 was quite easy. · 1 year, 7 months ago

3rd one is preety easy. Do u still need it? · 2 years, 7 months ago

Yup I'm still stuck till now. =_=" · 2 years, 7 months ago

Trying #5...

$\displaystyle \frac{bc\sqrt{a-2} + ca\sqrt[3]{b-6} + ab\sqrt[4]{c-12}}{abc}$

$= \displaystyle \frac{\sqrt{a-2}}{a} + \frac{\sqrt[3]{b-6}}{b} + \frac{\sqrt[4]{c-12}}{c}$

Let $$x = a-2, y = b-6, z = c-12$$ such that $$x,y,z \geq 0$$.

The expression becomes $= \displaystyle \frac{\sqrt{x}}{x+2} + \frac{\sqrt[3]{y}}{y+6} + \frac{\sqrt[4]{z}}{z+12}$

By AM-GM; $\displaystyle \frac{\sqrt{x}}{x+2} \leq \frac{\sqrt{x}}{2\sqrt{2x}} = \frac{1}{2\sqrt{2}}$

Similarly, $\displaystyle \frac{\sqrt[3]{y}}{y+3+3} \leq \frac{\sqrt[3]{y}}{3\sqrt[3]{y\times3^{2}}} = \frac{1}{3\times 3^{2/3}}$

$\displaystyle \frac{\sqrt[4]{z}}{z+4+4+4} \leq \frac{\sqrt[4]{z}}{4\sqrt[4]{z\times4^{3}}} = \frac{1}{4\times 4^{3/4}}$

Sum all these up and we get

$\displaystyle \frac{bc\sqrt{a-2} + ca\sqrt[3]{b-6} + ab\sqrt[4]{c-12}}{abc} \leq \frac{1}{2\sqrt{2}}+\frac{1}{3\times 3^{2/3}}+\frac{1}{4\times 4^{3/4}}$ ~~~

Equality occurs if and only if $$x=2,y=3,z=4$$ which means $$a=4,b=9,c=16$$ ~~~

I don't have much time for doing these problems right now. I have to study math from POSN for whole October and I'm freaking busy reviewing lots of stuffs. heavy breathing · 2 years, 7 months ago