I was doing the the following problem-

**Prove that**\( \frac { \sqrt { a+b+c } +\sqrt { a } }{ b+c } +\frac { \sqrt { a+b+c } +\sqrt { b } }{ c+a } +\frac { \sqrt { a+b+c } +\sqrt { c } }{ a+b } \ge \frac { 9+3\sqrt { 3 } }{ 2\sqrt { a+b+c } } \).

I normalized this to \(a+b+c=1\) and simplified to get-

\(\frac { 1 }{ 1-\sqrt { a } } +\frac { 1 }{ 1-\sqrt { b } } +\frac { 1 }{ 1-\sqrt { c } } \ge \frac { 9+3\sqrt { 3 } }{ 2 }\)

By Titu's lemma,

\(\Rightarrow \quad \frac { 1 }{ 1-\sqrt { a } } +\frac { 1 }{ 1-\sqrt { b } } +\frac { 1 }{ 1-\sqrt { c } } \ge \frac { 9 }{ 3-(\sqrt { a } +\sqrt { b } +\sqrt { c } ) } \)

However the \(RHS\) is maximized when \(\sqrt { a } +\sqrt { b } +\sqrt { c } \) is maximized which is at \(\sqrt { 3 } \).

\(\Rightarrow \quad \frac { 1 }{ 1-\sqrt { a } } +\frac { 1 }{ 1-\sqrt { b } } +\frac { 1 }{ 1-\sqrt { c } } \ge \frac { 9 }{ 3-(\sqrt { a } +\sqrt { b } +\sqrt { c } ) } \le \frac { 9 }{ 3-\sqrt { 3 } } =\frac { 9+3\sqrt { 3 } }{ 2 } \)

What went wrong?

You can view the rest of the problems here

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestHello Siddharth G,

Appeared in RMO 2015? How was it? What is your expected score?

Log in to reply

4 Questions. How was yours?

Log in to reply

Actually I am in K.V JNU so i cannot appear in RMO directly. I have to qualify JMO, which i gave this year but unfortunately did not make it. I will try next year( class 11th).

By the way, how was your preparation this year?

Did you study something more this year than what you studied last year for RMO 2014?

Log in to reply

Log in to reply

Log in to reply

Log in to reply

What is the your probability of clearing RMO?

Log in to reply

Log in to reply

As you have qualified RMO 2014, can you tell me that is the INMO camps conducted in delhi or no camp is conducted?

Log in to reply

Log in to reply

Which topics were taught to you?

If possible, can you send me some notes of that camp to my e-mail?

E-mail- priyanshu_2feb@rediffmail.com

Log in to reply

Log in to reply

Log in to reply

Have you solved this RMO question in the exam? :

Show that there are infinitely many triples \((x, y, z)\) of integers such that \(x^3 + y^4 = z^{31}\).

Log in to reply

Log in to reply

Before getting this, what steps you did?

Log in to reply

Log in to reply

Log in to reply

Log in to reply

I was doing this question:

Find all positive integers \(n\) such that \(3^{n - 1} + 5^{n - 1}\) divides \(3^n + 5^n\).

The solution in the book was:

Note that \(\large\ 1 < \frac { { 3 }^{ n } + { 5 }^{ n } }{ { 3 }^{ n - 1 } + { 5 }^{ n - 1 } } <5\), so we can have only \(\large\ \frac { { 3 }^{ n } + { 5 }^{ n } }{ { 3 }^{ n - 1 } + { 5 }^{ n - 1 } } \in \{ 2,3,4\}\)

cases, which are easily checked.

Can you explain me why the solution wrote \(\large\ 1 < \frac { { 3 }^{ n } + { 5 }^{ n } }{ { 3 }^{ n - 1 } + { 5 }^{ n - 1 } } <5\) above ?

I am not understanding how that expression came.

Log in to reply

Log in to reply

Why it chose 1 and 5 only?

Log in to reply

Log in to reply

Are the RMO results out?

Log in to reply

Log in to reply

Were they geometry ones?

Log in to reply

Log in to reply

Show that there are infinitely many positive real numbers \(a\) which are not integers such that \(a(a-3\text{{a}})\) is an integer.

Which method or theorem you applied?

Log in to reply

Log in to reply

I am unable to draw it even after trying 5 times?

Please elaborate the steps.

Log in to reply

Log in to reply

how you drew the image of the rmo question on the website which you sent me?

Please tell the website where I can draw the diagrams.

Log in to reply

Log in to reply

No I don't have solution to that inequality .

Please tell me how to save the diagrams of geogebra in desktop as soon as possible.

Log in to reply

Log in to reply

Should I need to login on the geogebra?

Log in to reply

Log in to reply

By the way how's your preparation for INMO? Was INMOTC conducted at IIT?

If possible, please send me the assignments. You can take as many days you can.

Log in to reply

Log in to reply

IIT professors don't take it initiative. They are just regional coordinators for name.

In KV, 20 DAYS camp is conducted but rarely a student is selected in INMO, where students from delhi region attend no camp, yet so many qualify. GREAT IRONY!!

Also thanks for helping me to post geometric figures. The method worked.

How was your INMO? Can you send me the paper?

Also, clear my doubts of 2 questions i have posted - "Help residues mod 7 continents".

Please post solutions also.

Log in to reply

Log in to reply

Log in to reply

Find all positive integers \((a, b)\) such that \(\large\ \frac { \sqrt { 2 } + \sqrt { a } }{ \sqrt { 3 } + \sqrt { b } }\) is a rational number.

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Also is it possible for you to send me the notes of INMOTC conducted in Delhi and Hyderabad?

Log in to reply

Log in to reply

I am presently in class 10. Yes, i appeared in JMO in 2015 but could not qualify as i scored just \(48\) marks out of 100.

If you cannot send the notes, then can you atleast send me the assignments provided during the camp?

You can take as many days you want.

Log in to reply

Log in to reply

By the way, which camp was good - DELHI or HYDERABAD?

Log in to reply

Log in to reply

Log in to reply

Log in to reply

This is the list of JMO 2015 qualifiers:

Log in to reply

Log in to reply

Log in to reply

Log in to reply

By the way, in which class are you?

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Can you tell me what is meant by the test "SMT" conducted in all DPSs? I got to know about it from my fiitjee friend who studies at DPS RKP.

Log in to reply

Log in to reply

Have you solved question 1 of INMO?

Also help me in these questions:

Log in to reply

Log in to reply

Also you share this problem with your friends of school and FiitJee.

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Hope that you qualify INMO 2016 also, with flying colours.

Log in to reply

Log in to reply

\(f(x)\) is a fifth degree polynomial. It is given that \(f(x) + 1\) is divisible by \((x - 1)^3\) and \(f(x) - 1\) is divisible by \((x + 1)^3\). Find \(f(x)\).

Log in to reply

Log in to reply

Log in to reply

Integration or differentiation?

Thanks, but I am unable to understand your solution.

Log in to reply

Log in to reply

According to my knowledge we can differentiate a function only w.r.t some variable , but here are 6 variables.

Could you elaborate the differentiation here?

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Sorry for your inconvenience.

Log in to reply

Log in to reply

Can you tell me how to start?

Also thanks a lot for sending INMO 2015 notes.

Log in to reply

Log in to reply

Log in to reply

Log in to reply

I have found another solution without calculus(working half an hour on it). It is completely algebraic.

Here it is:

Let \(f(x) = (x - 1)^3(a{x}^2 + bx + c) - 1\). \(...(1)\)

As \((x + 1)^3|f(x) - 1\) we have

\(f(x) - 1 = a{x}^5 + x^4(b - 3a) + x^3(c - 3b + 3a) + x^2(3b - 3c - a) + x(3c - b) - c - 2\).

and the RHS is divisible by \((x + 1)^3\) , so by dividing RHS by \((x + 1)^3\)

We have the remainder \((-38a - 6c +18b)x^2 + (-48a + 16b)x - 2c - 18a +6b - 2 = 0\)

So, we have three simultaneous equations viz:

\(38a + 6c -18b = 0\) ; \(48a - 16b = 0\) ; \(2c + 18a - 6b + 2 = 0\)

which on solving together gives \(a = \frac {-3}{8}\) ; \(b = \frac {-9}{8}\) ; \(c = -1\).

Putting these values in \((1)\) and doing some tedious calculations we get

\(f(x) = \boxed{\frac { -3 }{ 8 } { x }^{ 5 }+ \frac { 5 }{ 4 } { x }^{ 3 } - \frac { 15 }{ 8 } x}\) which is the correct answer.

Can you please rate my solution?

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

The arithmetic mean of a pair wise distinct prime numbers is 27. Determine the biggest prime among them.

Log in to reply

Log in to reply

Which inequality will you apply to solve this ?:

a, b, c are real numbers such that their sum is \(0\) an sum of their square is 1. Find the maximum value of \(a^2.b^2.c^2\).

Log in to reply

Log in to reply

Log in to reply

Log in to reply

When will you send the assignments to my mail?

Log in to reply

Nice solution!

The second last line should be; "RHS is maximised when √a+ √b+ √c is 'minimised' " as the whole fraction is maximised when its denominator is minimised.

However your deduction is very perfect.

In which class presently you are?

Log in to reply

11th As for the second last line, \(\sqrt{a} +\sqrt{b} +\sqrt{c}\) when maximized, yields \(3-(\sqrt{a} +\sqrt{b} +\sqrt{c})\) to be minimized,which implies that the RHS is maximized.

Log in to reply

Thanks. Now i understood that.

I am also preparing for RMO. So as an elder please guide me that which book will be helpful for me to clear RMO?

Also i get to know that you have cleared RMO , so please tell me which book helped you a lot.

Log in to reply

Log in to reply

Are 'Problem primer for the olympiads' and 'Number theory-problems, structures and examples'(by titu .A.) not fine for RMO/INMO ?

Log in to reply

Log in to reply

Please help me in solving this question ( KVS JMO 2015)

A polynomial \(f(x)\) with rational coefficients leaves remainder \(15\), when divided by \(x-3\) and remainder \(2x+1\), when divided by \((x-1)^2\). Find the remainder when \(f(x)\) is divided by \((x-3)(x-1)^2\).

Log in to reply

Log in to reply

You can tell me your approach, irrespective of good or bad.

Log in to reply

\(2x^3-4x^2+4x+1 = (x-1)^2(x+2) + 2x+1\). It left a remainder

31when divided by \(x-3\). Note that \((x-1)^2\) leaves a remainder of4when divided by \(x-3\).Thus, \(2x^3-4x^2+4x+1 - 4(x-1)^2 =2x^3-8x^2+12x-3\) leaves a remainder of \(31-4(4)=15\). Hence we find the remainder of \(2x^3-8x^2+12x-3\) when divided by \((x-1)^2(x-3)\).

Log in to reply

Then, \(f(x)=q_1(x)(x-3)(x-1)^2+r_1(x)\), where \(r_1(x)=ax^2+bx+c\).

Dividing by \((x-1)^2\) and equating remainders,

\(2x+1=0+(ax^2+bx+c-a(x-1)^2)\) [Multiplied by

ato remove the coeff. of \(x^2\)]\(\Rightarrow 2x+1=(2a-b)x+(c-a)\) \(\Rightarrow 2=2a-b, 1=c-a\)

Solving with \(f(1)=3=a+b+c -(1)\)

(a=2, b=-2, c=3)

Log in to reply

I have your 2nd solution.

What is the problem that i am not able to understand that why you have multiplied x with q1 ( 2nd line) in the polynomial?

You have taken the polynomial as

\(f(x)\quad =\quad { q }_{ 1 }(x)(x-3){ (x-1) }^{ 2 }\quad +\quad { r }_{ 1 }(x)\)

I am asking that why you have multiplied x in above polynomial as it is not given?

Log in to reply

Log in to reply

Also give me a clue to solve this problem:

Find all real numbers \(x\) for which

\({ 10 }^{ x }\quad +\quad { 11 }^{ x }\quad +\quad { 12 }^{ x }\quad =\quad { 13 }^{ x }\quad +\quad { 14 }^{ x }\)

Log in to reply

Log in to reply

Please tell me that can i apply CAUCHY - SCHWARZ lemma to prove this inequality ? -

Given \(a, b, c\) are positive real numbers such that

\({ a }^{ 2 }\quad +\quad { b }^{ 2 }\quad +\quad { c }^{ 2 }\quad =\quad 3abc\).

Prove that:

\(\large\ \frac { a }{ { b }^{ 2 }{ c }^{ 2 } } +\frac { b }{ { c }^{ 2 }{ a }^{ 2 } } +\frac { c }{ { a }^{ 2 }{ b }^{ 2 } } \ge \quad \frac { 9 }{ a+b+c }\)

Log in to reply

Log in to reply

As you have cleared RMO, can you tell me which book you preferred specially for Geometry and functional equations?

Log in to reply

Log in to reply

Any professor or you did by yourself or anyone else?

Also without geometry how can one dream for winning RMO or further olympiads? It is the whole and sole .

Log in to reply

Log in to reply

Please help me to factorise this one:

\(\large\ { x }^{ 2 }+4{ y }^{ 2 }-2xy-2x-4y-8=0\)

Log in to reply

Log in to reply

Log in to reply

Log in to reply

\(\large\ \frac { { a }^{ 4 } }{ { a }^{ 3 }{ b }^{ 2 }{ c }^{ 2 } } +\frac { { b }^{ 4 } }{ { b }^{ 3 }{ c }^{ 2 }{ a }^{ 2 } } +\frac { { c }^{ 4 } }{ { c }^{ 3 }{ a }^{ 2 }{ b }^{ 2 } } \ge \frac { 9 }{ a+b+c }\)

\(\large\ \frac { { a }^{ 4 } }{ { a }^{ 3 }{ b }^{ 2 }{ c }^{ 2 } } +\frac { { b }^{ 4 } }{ { b }^{ 3 }{ c }^{ 2 }{ a }^{ 2 } } +\frac { { c }^{ 4 } }{ { c }^{ 3 }{ a }^{ 2 }{ b }^{ 2 } } \ge \frac { { ({ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }) }^{ 2 } }{ { a }^{ 2 }{ b }^{ 2 }{ c }^{ 2 }(a+b+c) }\)

\(\large\ \quad \ge \frac { { 9({ abc }) }^{ 2 } }{ { a }^{ 2 }{ b }^{ 2 }{ c }^{ 2 }(a+b+c) }\)

\(\large\ \quad =\frac { 9 }{ a+b+c }\)

Log in to reply

@Siddharth G :- many congrats for clearing RMO!!!. How many problems did you solve?

Log in to reply

Thanks! I attempted 5 questions, expected to get 4-4.5. How was your paper? (GMO right?)

Log in to reply

Read This. I should say, I didn't do as well as you did. (Regardless of the paper's standard)

Log in to reply