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# A doubt!

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

Note by Anik Mandal
7 months, 1 week ago

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· 7 months ago

Great! A solution through classical inequalities should be better. · 7 months ago

I Prefer Trigonometry . Also i tried to use cauchy, titu but couldn't succeed therefore i switched to trigonometric substitution · 7 months ago

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

Can anyone post an algebraic proof? · 7 months ago

How's the proof? · 7 months ago

hey @Anik Mandal i got the result! · 7 months ago

Looks like an application of Cauchy's/Titu's · 7 months, 1 week ago

Did you get the result? · 7 months, 1 week ago

@Sharky Kesa · 7 months, 1 week ago

Setting xy=t maybe fruitful, but I have not tried it. · 7 months, 1 week ago

Are you sure the problem is correct, because x=0.5 and y=$$-0.5$$ are not satisfying the condition. · 7 months, 1 week ago

They are not satisfying first condition only. · 7 months, 1 week ago

Oh sorry I misread x^2 +y^2 as x+y. · 7 months, 1 week ago

I believe this was an RMO problem of some year.Anyways you'll have your RMO this Sunday right? · 7 months, 1 week 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. · 7 months, 1 week ago

Ok!Are you there on Slack or hangouts?I mean are you active? · 7 months, 1 week ago

Yeah, wbu? · 7 months, 1 week ago

Same. · 7 months, 1 week ago

@Harsh Shrivastava Can you help? · 7 months, 1 week ago

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 $$0 \leq xy \leq \dfrac{1}{2}$$. · 7 months ago

I Have sent you that integral on mail. check it out! · 6 months, 3 weeks ago

Very Slick! +1. i was expecting an algebraic solution from you · 7 months ago

thanks !. nice use of trig substitution by the way. · 7 months ago

actually I misread the question.fhe proof Only Works For Positive x And y. · 7 months ago

Ohh!. i didn't noticed. BTW Thanks! :) · 6 months, 4 weeks ago

Is gravitation important for kvpy?

(Sorry for asking this at wrong place.) · 6 months, 4 weeks ago

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! :) · 6 months, 4 weeks ago

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

Any further tips?

Thanks. · 6 months, 4 weeks ago

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! · 6 months, 4 weeks ago

Thanks for the tip. · 6 months, 4 weeks ago

You can ask @rohit kumar also.he also qualified it · 6 months, 4 weeks ago