# 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
3 years, 9 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 $0 \leq xy \leq \dfrac{1}{2}$.

- 3 years, 9 months ago

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

- 3 years, 9 months ago

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

- 3 years, 9 months ago

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

- 3 years, 9 months ago

Is gravitation important for kvpy?

(Sorry for asking this at wrong place.)

- 3 years, 9 months 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! :)

- 3 years, 9 months ago

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

Any further tips?

Thanks.

- 3 years, 9 months 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!

- 3 years, 9 months ago

Thanks for the tip.

- 3 years, 9 months ago

You can ask @rohit kumar also.he also qualified it

- 3 years, 9 months ago

My pleasure!. do well :)

- 3 years, 9 months ago

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

- 3 years, 9 months ago

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

- 3 years, 9 months ago

How's the proof?

- 3 years, 9 months ago IMAGE IMAGE

- 3 years, 9 months ago

Great! A solution through classical inequalities should be better.

- 3 years, 9 months ago

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

- 3 years, 9 months ago

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

Can anyone post an algebraic proof?

- 3 years, 9 months ago

hey @Anik Mandal i got the result!

- 3 years, 9 months ago

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

- 3 years, 9 months ago

Did you get the result?

- 3 years, 9 months ago

- 3 years, 9 months ago

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

- 3 years, 9 months ago

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

- 3 years, 9 months ago

They are not satisfying first condition only.

- 3 years, 9 months ago

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

- 3 years, 9 months ago

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

- 3 years, 9 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.

- 3 years, 9 months ago

Ok!Are you there on Slack or hangouts?I mean are you active?

- 3 years, 9 months ago

Yeah, wbu?

- 3 years, 9 months ago

Same.

- 3 years, 9 months ago

@Harsh Shrivastava Can you help?

- 3 years, 9 months ago