Prove the following inequality

ab+2c+3d+bc+2d+3a+cd+2a+3b+da+2b+3c>23\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.

Note by Raushan Sharma
3 years, 9 months ago

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Have u given the inmo mock test of fiitjee yesterday

Samarth Agarwal - 3 years, 9 months ago

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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 - 3 years, 9 months ago

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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 - 3 years, 9 months ago

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@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 - 3 years, 9 months ago

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@Raushan Sharma Use titu lemma and then cauchy schwarz on sqrt(a),sqrt(b),sqrt(c),sqrt(d)and1,1,1,1sqrt (a), sqrt (b), sqrt (c), sqrt (d) and 1,1,1,1 it would give the direct result

Samarth Agarwal - 3 years, 9 months ago

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@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(a+b+c+d)^2. I first applied Tittu's lemma and then AM-GM

Raushan Sharma - 3 years, 9 months ago

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

Saurabh Mallik - 3 years, 7 months ago

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

Raushan Sharma - 3 years, 7 months ago

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@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 - 3 years, 7 months ago

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

cimo DD - 2 months, 1 week ago

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