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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## Comments

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Great! A solution through classical inequalities should be better.

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

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

Can anyone post an algebraic proof?

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@Harsh Shrivastava Can you help?

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Are you sure the problem is correct, because x=0.5 and y=\(-0.5\) are not satisfying the condition.

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They are not satisfying first condition only.

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Oh sorry I misread x^2 +y^2 as x+y.

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Setting xy=t maybe fruitful, but I have not tried it.

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@Sharky Kesa

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Looks like an application of Cauchy's/Titu's

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Did you get the result?

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hey @Anik Mandal i got the result!

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@Anik Mandal @Harsh Shrivastava @Sharky Kesa

How's the proof?

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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 \( 0 \leq xy \leq \dfrac{1}{2} \).

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Very Slick! +1. i was expecting an algebraic solution from you

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thanks !. nice use of trig substitution by the way.

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I Have sent you that integral on mail. check it out!

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actually I misread the question.fhe proof Only Works For Positive x And y.

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Ohh!. i didn't noticed. BTW Thanks! :)

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(Sorry for asking this at wrong place.)

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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! :)

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Any further tips?

Thanks.

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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!

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@rohit kumar also.he also qualified it

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