Prove the following inequality

$\dfrac{a}{b+2c+3d} + \dfrac{b}{c+2d+3a} + \dfrac{c}{d+2a+3b} + \dfrac{d}{a+2b+3c} > \dfrac{2}{3}$

If a,b,c,d are distinct positive reals, prove the above inequality.

#Algebra

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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}`	$\boxed{123}$

Comments

Have u given the inmo mock test of fiitjee yesterday

Samarth Agarwal - 5 years, 5 months ago

No, one of my friends gave that, and yeah, it's a question I gave from there only... Did you give that test?? How was it?

Raushan Sharma - 5 years, 5 months ago

Yes I gave the test and it was really difficult....this was the only question I could completely solve....I would post the complete paper soon.

Samarth Agarwal - 5 years, 5 months ago

@Samarth Agarwal – Ya, I have the complete paper, but can you give the solution to this inequality, I mean how you solved, I was trying it with Tittu's Lemma, but couldn't complete

Raushan Sharma - 5 years, 5 months ago

@Raushan Sharma – Use titu lemma and then cauchy schwarz on $sqrt (a), sqrt (b), sqrt (c), sqrt (d) and 1,1,1,1$ it would give the direct result

Samarth Agarwal - 5 years, 5 months ago

@Samarth Agarwal – Oh, yeah, I did it today, after I commented that. It was quite easy. Actually first I was not expanding $(a+b+c+d)^2$ . I first applied Tittu's lemma and then AM-GM

Raushan Sharma - 5 years, 5 months ago

@Raushan Sharma – Can you please post the complete solution including explanation for this question??

Saurabh Mallik - 5 years, 3 months ago

@Saurabh Mallik – Yes, I can, but for that can you please tell me, how can we add an image in the comment??

Raushan Sharma - 5 years, 3 months ago

@Raushan Sharma – Sorry. I don't know how can we upload images in comment. I think we can't upload images in comment but and upload it in solutions for given questions. Can you please write the whole solution?

Saurabh Mallik - 5 years, 3 months ago

Quite easy cyclic sum(a^2/(ab+2ac+3ad)) and apply Tittu's lemma

cimo DD - 1 year, 10 months 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}`	$\boxed{123}$