Let \(x\) and \(y\) be real numbers such that \(x^2+y^2=1\) .Prove that \[\dfrac{1}{1+x^2}+ \dfrac{1}{1+y^2} + \dfrac{1}{1+xy} \geq \frac{3}{1+(\dfrac{x+y}{2})^2}\]

Please help me this as soon as possible.Thanks!

Let \(x\) and \(y\) be real numbers such that \(x^2+y^2=1\) .Prove that \[\dfrac{1}{1+x^2}+ \dfrac{1}{1+y^2} + \dfrac{1}{1+xy} \geq \frac{3}{1+(\dfrac{x+y}{2})^2}\]

Please help me this as soon as possible.Thanks!

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– Anik Mandal · 2 months, 3 weeks ago

Great! A solution through classical inequalities should be better.Log in to reply

– Prakhar Bindal · 2 months, 3 weeks ago

I Prefer Trigonometry . Also i tried to use cauchy, titu but couldn't succeed therefore i switched to trigonometric substitutionLog in to reply

@rohit kumar @Aniket Sanghi @ARYAN GOYAT @Archit Agrawal @Aditya Chauhan

Can anyone post an algebraic proof? – Prakhar Bindal · 2 months, 3 weeks ago

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@Anik Mandal @Harsh Shrivastava @Sharky Kesa

How's the proof? – Prakhar Bindal · 2 months, 3 weeks ago

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hey @Anik Mandal i got the result! – Prakhar Bindal · 2 months, 3 weeks ago

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Looks like an application of Cauchy's/Titu's – Sharky Kesa · 3 months ago

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– Anik Mandal · 3 months ago

Did you get the result?Log in to reply

@Sharky Kesa – Anik Mandal · 3 months ago

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Setting xy=t maybe fruitful, but I have not tried it. – Harsh Shrivastava · 3 months ago

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Are you sure the problem is correct, because x=0.5 and y=\(-0.5\) are not satisfying the condition. – Harsh Shrivastava · 3 months ago

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– Anik Mandal · 3 months ago

They are not satisfying first condition only.Log in to reply

– Harsh Shrivastava · 3 months ago

Oh sorry I misread x^2 +y^2 as x+y.Log in to reply

– Anik Mandal · 3 months ago

I believe this was an RMO problem of some year.Anyways you'll have your RMO this Sunday right?Log in to reply

– Harsh Shrivastava · 3 months ago

Bro I'll try this problem tonight because I have to goto Fiitjee after some time, and need to study chemistry, The problem seems to be tricky.Log in to reply

– Anik Mandal · 3 months ago

Ok!Are you there on Slack or hangouts?I mean are you active?Log in to reply

– Harsh Shrivastava · 3 months ago

Yeah, wbu?Log in to reply

– Anik Mandal · 3 months ago

Same.Log in to reply

@Harsh Shrivastava Can you help? – Anik Mandal · 3 months ago

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First I would like to prove the following: \(\dfrac{1}{1+x^2} + \dfrac{1}{1+y^2} \geq \dfrac{2}{1+{\frac{(x+y)}{2}}^{2}} \dots (1) \) for \( \dfrac{1}{4} \leq xy \leq \dfrac{1}{2} \)

\( \dfrac{2 + x^2 + y^2}{1 + x^2 + y^2 + {x}^{2}{y}^{2}} \geq \dfrac{2}{1 + \frac{1+2xy}{4}} \)

\( \dfrac{3}{2+{x}^{2}{y}^{2}} \geq \dfrac{2}{\frac{5}{4} + \frac{xy}{2}} \)

\( 8{xy}^{2} -6xy +1 \leq 0 \)

This is true for \( \frac{1}{4} \leq xy \leq \frac{1}{2} \). Thus it has been proved.

\( \dfrac{1}{1+xy} \geq \dfrac{1}{1+{\frac{(x+y)}{4}}^{2}} \dots (2) \) (trivial)

Now add (1) and (2).Thus the result is proved for \( \dfrac{1}{4} \leq xy \leq \dfrac{1}{2} \).

What remains (\( 0 \leq xy \leq \frac{1}{4} \)) is simple.

The minimum value of the L.H.S in this interval is \( \dfrac{16}{11} + \dfrac{5}{4} \). The maximum value of the R.H.S is \( \dfrac{12}{5} \).

This proves the result for \( 0 \leq xy \leq \dfrac{1}{2} \). – Rohit Kumar · 2 months, 3 weeks ago

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– Prakhar Bindal · 2 months, 3 weeks ago

I Have sent you that integral on mail. check it out!Log in to reply

– Prakhar Bindal · 2 months, 3 weeks ago

Very Slick! +1. i was expecting an algebraic solution from youLog in to reply

– Rohit Kumar · 2 months, 3 weeks ago

thanks !. nice use of trig substitution by the way.Log in to reply

– Rohit Kumar · 2 months, 3 weeks ago

actually I misread the question.fhe proof Only Works For Positive x And y.Log in to reply

– Prakhar Bindal · 2 months, 3 weeks ago

Ohh!. i didn't noticed. BTW Thanks! :)Log in to reply

(Sorry for asking this at wrong place.) – Harsh Shrivastava · 2 months, 3 weeks ago

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But Last year KVPY Math and chem were quite easy. Bio was a nightmare(atleast for me)

My physics didn't went that well as it had some problems from optics which wasn't taught at that point of time.

But yeah if you are appearing for NSEA Then do master Gravitation!!.

For KVPY Do study current electricity , optics in physics .

Hydrocarbons in chemistry and basics taught in class 10th (Mensuration , Volume and surface area , Elementary Number theory) that will be enought

And Yeah Geometry is also important.

All the best! :) – Prakhar Bindal · 2 months, 3 weeks ago

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Any further tips?

Thanks. – Harsh Shrivastava · 2 months, 3 weeks ago

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chem is easy but they ask some questions of organic chemistry which i suppose is there in phase-3.

Studying NCERT Will be enough for that

physics will be easy of the topics you have been taught.

All the best! – Prakhar Bindal · 2 months, 3 weeks ago

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– Harsh Shrivastava · 2 months, 3 weeks ago

Thanks for the tip.Log in to reply

@rohit kumar also.he also qualified it – Prakhar Bindal · 2 months, 3 weeks ago

You can askLog in to reply

– Prakhar Bindal · 2 months, 3 weeks ago

My pleasure!. do well :)Log in to reply