Need help with inequality!!!

This problem seems so simple, but I cannot prove it:

$a,b,c>0$ .Prove that:

$(a^2+2)(b^2+2)(c^2+2)\geq 9(ab+bc+ca)$

I tried all sorts of inequalities, but they didn't work.I reduced it to proving

$(x+2)(y+2)(z+2)\geq9(x+y+z)$ , but I cannot prove that neither.

#Algebra #HelpMe! #Inequality #Pleasehelp

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Comments

Hi Bogdan, This question is from APMO exam. You can find the solutions here But, I can add a another solution with some trignometry here. We choose $A,B,C$ such that $a=\sqrt{2}tanA$ , $b=\sqrt{2}tanB$ , $c=\sqrt{2}tanC$ . Now, by using $1+tan^{2}x=\frac{1}{cos^2x}$ , We get $\frac{4}{9}≥cosAcosBcosC(cosAsinBsinC+sinAcosBsinC+sinAsinBcosC)$ , Which is same as proving $\frac{4}{9}≥cosAcosBcosC(cosAcosBcosC-cos(A+B+C))$ . Now, we have some "BAD LOOKING" Trignometric terms. So lets find a way to cancel them out. OK, then letting $y=A+B+C$ . Now bu Simple AM-GM, We Get $cosAcosBcosC≤(\frac{cosA+cosB+cosC}{3})^3≤cos^3y$ . Thus, Finally, we are left to show that $\frac{4}{9}≥cos^3y(cos^3y-cos(3y))$ . Now, by very easy simplification , we need to show $\frac{4}{27}≥cos^4y(1-cos^2y)$ , Which follows from simple AM-GM As $(\frac{cos^2y}{2}.\frac{cos^2y}{2}.(1-cos^2y))^{\frac{1}{3}}≤\frac{1}{3}(\frac{cos^2y}{2}+\frac{cos^2y}{2}+(1-cos^2y))=\frac{1}{3}$ . And the equality holds here at $a=b=c=1$

Dinesh Chavan - 7 years ago

Thanks for the quick response!

Bogdan Simeonov - 7 years ago

Ok, Hope you might have cleared ur doubts

Dinesh Chavan - 7 years ago

@Dinesh Chavan – Why are your comments down voted?Also, at the end of the discussion someone mentioned a pretty strong result and didn't prove it.Any ideas?

Bogdan Simeonov - 7 years ago

@Bogdan Simeonov – Wait, I will try now,, For the downvotes, I know who did it, Nevermind :)

Dinesh Chavan - 7 years ago

@Dinesh Chavan – Ok :D .I'll try it also.

Bogdan Simeonov - 7 years ago

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}`	$\boxed{123}$