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A doubt!

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!

Note by Anik Mandal
9 months, 1 week ago

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Prakhar Bindal · 9 months ago

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@Prakhar Bindal Great! A solution through classical inequalities should be better. Anik Mandal · 9 months ago

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@Anik Mandal I Prefer Trigonometry . Also i tried to use cauchy, titu but couldn't succeed therefore i switched to trigonometric substitution Prakhar Bindal · 9 months ago

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@Anik Mandal @Harsh Shrivastava @Sharky Kesa

How's the proof? Prakhar Bindal · 9 months ago

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hey @Anik Mandal i got the result! Prakhar Bindal · 9 months ago

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Looks like an application of Cauchy's/Titu's Sharky Kesa · 9 months, 1 week ago

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@Sharky Kesa Did you get the result? Anik Mandal · 9 months, 1 week ago

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@Sharky Kesa Anik Mandal · 9 months, 1 week ago

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Setting xy=t maybe fruitful, but I have not tried it. Harsh Shrivastava · 9 months, 1 week 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 · 9 months, 1 week ago

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@Harsh Shrivastava They are not satisfying first condition only. Anik Mandal · 9 months, 1 week ago

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@Anik Mandal Oh sorry I misread x^2 +y^2 as x+y. Harsh Shrivastava · 9 months, 1 week ago

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@Harsh Shrivastava I believe this was an RMO problem of some year.Anyways you'll have your RMO this Sunday right? Anik Mandal · 9 months, 1 week ago

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@Anik Mandal 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. Harsh Shrivastava · 9 months, 1 week ago

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@Harsh Shrivastava Ok!Are you there on Slack or hangouts?I mean are you active? Anik Mandal · 9 months, 1 week ago

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@Anik Mandal Yeah, wbu? Harsh Shrivastava · 9 months, 1 week ago

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@Harsh Shrivastava Same. Anik Mandal · 9 months, 1 week ago

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@Harsh Shrivastava Can you help? Anik Mandal · 9 months, 1 week 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 · 9 months ago

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@Rohit Kumar I Have sent you that integral on mail. check it out! Prakhar Bindal · 8 months, 3 weeks ago

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@Rohit Kumar Very Slick! +1. i was expecting an algebraic solution from you Prakhar Bindal · 9 months ago

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@Prakhar Bindal thanks !. nice use of trig substitution by the way. Rohit Kumar · 9 months ago

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@Rohit Kumar actually I misread the question.fhe proof Only Works For Positive x And y. Rohit Kumar · 9 months ago

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@Rohit Kumar Ohh!. i didn't noticed. BTW Thanks! :) Prakhar Bindal · 9 months ago

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@Prakhar Bindal Is gravitation important for kvpy?

(Sorry for asking this at wrong place.) Harsh Shrivastava · 9 months ago

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@Harsh Shrivastava i cant tell .But its very very important from NSEA

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 · 9 months ago

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@Prakhar Bindal I have not given any special focus to kvpy preparation until now, so is it late?

Any further tips?

Thanks. Harsh Shrivastava · 9 months ago

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@Harsh Shrivastava Study bio of class 10 if you want a very good rank . My rank would have been lot better if my score in bio was good(it was about 7/25 in aptitude test). Math would be easy for you .

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 · 9 months ago

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@Prakhar Bindal Thanks for the tip. Harsh Shrivastava · 9 months ago

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@Harsh Shrivastava You can ask @rohit kumar also.he also qualified it Prakhar Bindal · 9 months ago

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@Harsh Shrivastava My pleasure!. do well :) Prakhar Bindal · 9 months ago

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