A doubt!

Let xx and yy be real numbers such that x2+y2=1x^2+y^2=1 .Prove that 11+x2+11+y2+11+xy31+(x+y2)2\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
3 years, 1 month ago

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1 vote

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Prakhar Bindal - 3 years, 1 month ago

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Great! A solution through classical inequalities should be better.

Anik Mandal - 3 years, 1 month ago

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

Prakhar Bindal - 3 years, 1 month ago

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

Can anyone post an algebraic proof?

Prakhar Bindal - 3 years, 1 month ago

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@Harsh Shrivastava Can you help?

Anik Mandal - 3 years, 1 month ago

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

Harsh Shrivastava - 3 years, 1 month ago

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They are not satisfying first condition only.

Anik Mandal - 3 years, 1 month ago

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

Harsh Shrivastava - 3 years, 1 month 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 - 3 years, 1 month ago

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@Anik Mandal Yeah, wbu?

Harsh Shrivastava - 3 years, 1 month ago

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@Harsh Shrivastava Same.

Anik Mandal - 3 years, 1 month 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 - 3 years, 1 month ago

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

Anik Mandal - 3 years, 1 month ago

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

Harsh Shrivastava - 3 years, 1 month ago

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@Sharky Kesa

Anik Mandal - 3 years, 1 month ago

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

Sharky Kesa - 3 years, 1 month ago

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Did you get the result?

Anik Mandal - 3 years, 1 month ago

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hey @Anik Mandal i got the result!

Prakhar Bindal - 3 years, 1 month ago

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@Anik Mandal @Harsh Shrivastava @Sharky Kesa

How's the proof?

Prakhar Bindal - 3 years, 1 month ago

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First I would like to prove the following: 11+x2+11+y221+(x+y)22(1)\dfrac{1}{1+x^2} + \dfrac{1}{1+y^2} \geq \dfrac{2}{1+{\frac{(x+y)}{2}}^{2}} \dots (1) for 14xy12 \dfrac{1}{4} \leq xy \leq \dfrac{1}{2}

2+x2+y21+x2+y2+x2y221+1+2xy4 \dfrac{2 + x^2 + y^2}{1 + x^2 + y^2 + {x}^{2}{y}^{2}} \geq \dfrac{2}{1 + \frac{1+2xy}{4}}

32+x2y2254+xy2 \dfrac{3}{2+{x}^{2}{y}^{2}} \geq \dfrac{2}{\frac{5}{4} + \frac{xy}{2}}

8xy26xy+10 8{xy}^{2} -6xy +1 \leq 0

This is true for 14xy12 \frac{1}{4} \leq xy \leq \frac{1}{2} . Thus it has been proved.

11+xy11+(x+y)42(2) \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 14xy12 \dfrac{1}{4} \leq xy \leq \dfrac{1}{2} .

What remains (0xy14 0 \leq xy \leq \frac{1}{4} ) is simple.

The minimum value of the L.H.S in this interval is 1611+54 \dfrac{16}{11} + \dfrac{5}{4} . The maximum value of the R.H.S is 125 \dfrac{12}{5} .

This proves the result for 0xy12 0 \leq xy \leq \dfrac{1}{2} .

Rohit Kumar - 3 years, 1 month ago

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

Prakhar Bindal - 3 years, 1 month ago

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

Rohit Kumar - 3 years, 1 month ago

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

Prakhar Bindal - 3 years, 1 month ago

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

Rohit Kumar - 3 years, 1 month ago

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

Prakhar Bindal - 3 years, 1 month ago

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@Prakhar Bindal Is gravitation important for kvpy?

(Sorry for asking this at wrong place.)

Harsh Shrivastava - 3 years, 1 month 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 - 3 years, 1 month 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 - 3 years, 1 month 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 - 3 years, 1 month ago

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@Prakhar Bindal Thanks for the tip.

Harsh Shrivastava - 3 years, 1 month ago

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@Harsh Shrivastava My pleasure!. do well :)

Prakhar Bindal - 3 years, 1 month ago

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

Prakhar Bindal - 3 years, 1 month ago

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