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!

Easy Math Editor

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Comments

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Prakhar Bindal - 4 years, 7 months ago

Great! A solution through classical inequalities should be better.

Anik Mandal - 4 years, 7 months ago

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

Prakhar Bindal - 4 years, 7 months ago

@Prakhar Bindal – @rohit kumar @Aniket Sanghi @ARYAN GOYAT @Archit Agrawal @Aditya Chauhan

Can anyone post an algebraic proof?

Prakhar Bindal - 4 years, 7 months ago

@Harsh Shrivastava Can you help?

Anik Mandal - 4 years, 8 months ago

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

Harsh Shrivastava - 4 years, 8 months ago

They are not satisfying first condition only.

Anik Mandal - 4 years, 8 months ago

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

Harsh Shrivastava - 4 years, 8 months ago

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

Anik Mandal - 4 years, 8 months ago

@Anik Mandal – Yeah, wbu?

Harsh Shrivastava - 4 years, 8 months ago

@Harsh Shrivastava – Same.

Anik Mandal - 4 years, 8 months ago

@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 - 4 years, 8 months ago

@Harsh Shrivastava – Ok!Are you there on Slack or hangouts?I mean are you active?

Anik Mandal - 4 years, 8 months ago

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

Harsh Shrivastava - 4 years, 8 months ago

@Sharky Kesa

Anik Mandal - 4 years, 8 months ago

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

Sharky Kesa - 4 years, 8 months ago

Did you get the result?

Anik Mandal - 4 years, 8 months ago

hey @Anik Mandal i got the result!

Prakhar Bindal - 4 years, 7 months ago

@Anik Mandal @Harsh Shrivastava @Sharky Kesa

How's the proof?

Prakhar Bindal - 4 years, 7 months 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 $0 \leq xy \leq \dfrac{1}{2}$ .

Rohit Kumar - 4 years, 7 months ago

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

Prakhar Bindal - 4 years, 7 months ago

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

Rohit Kumar - 4 years, 7 months ago

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

Prakhar Bindal - 4 years, 7 months ago

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

Rohit Kumar - 4 years, 7 months ago

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

Prakhar Bindal - 4 years, 7 months ago

@Prakhar Bindal – Is gravitation important for kvpy?

(Sorry for asking this at wrong place.)

Harsh Shrivastava - 4 years, 7 months ago

@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 - 4 years, 7 months ago

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

Any further tips?

Thanks.

Harsh Shrivastava - 4 years, 7 months ago

@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 - 4 years, 7 months ago

@Prakhar Bindal – Thanks for the tip.

Harsh Shrivastava - 4 years, 7 months ago

@Harsh Shrivastava – My pleasure!. do well :)

Prakhar Bindal - 4 years, 7 months ago

@Harsh Shrivastava – You can ask @rohit kumar also.he also qualified it

Prakhar Bindal - 4 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$