# 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, 1 month ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $\boxed{123}$

Sort by: IMAGE IMAGE

- 3 years ago

Great! A solution through classical inequalities should be better.

- 3 years ago

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

- 3 years ago

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

Can anyone post an algebraic proof?

- 3 years ago

@Harsh Shrivastava Can you help?

- 3 years, 1 month ago

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

- 3 years, 1 month ago

They are not satisfying first condition only.

- 3 years, 1 month ago

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

- 3 years, 1 month ago

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

- 3 years, 1 month ago

Yeah, wbu?

- 3 years, 1 month ago

Same.

- 3 years, 1 month 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, 1 month ago

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

- 3 years, 1 month ago

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

- 3 years, 1 month ago

- 3 years, 1 month ago

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

- 3 years, 1 month ago

Did you get the result?

- 3 years, 1 month ago

hey @Anik Mandal i got the result!

- 3 years ago

How's the proof?

- 3 years 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 $0 \leq xy \leq \dfrac{1}{2}$.

- 3 years ago

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

- 3 years ago

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

- 3 years ago

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

- 3 years ago

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

- 3 years ago

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

- 3 years ago

Is gravitation important for kvpy?

(Sorry for asking this at wrong place.)

- 3 years 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 ago

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

Any further tips?

Thanks.

- 3 years 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 ago

Thanks for the tip.

- 3 years ago

My pleasure!. do well :)

- 3 years ago

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

- 3 years ago