Hi guys!

I know that many of you must be RMO aspirants, and are preparing tough for that. But even all of us know that RMO is not that easy to qualify.There are a lot of problems to do and concepts to learn.So why not discuss and gain more and more knowledge?

This board has been made for that purpose alone!

Please do share problems and concepts in this board, and ask uncountable number of doubts. Also discuss about books which can be helpful for RMO preparation. Some of them I recommend are :

Challenge and Thrills of Pre College Mathematics

Problem Solving Strategies by Arthur Engel

RMO and INMO book of Arihant Publication by Rajeev Manocha

Miscellaneous

Before going to prove stuff yourself, be aware of the basic proofs

Please do share *Concepts of the Day* and also the problems related to it. Do link question papers so that all of us can do them together. I hope the members of our community would be able to represent their respective countries in the **IMO**!

## Comments

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TopNewestGreat step!

Well, there are many who would be interested in this discussion.The toughest I feel is the number theory part of RMO. What are some of the good sources to prepare for it.

There are some exceptionally brilliant people on Brilliant who have the experience of RMO,INMO,IMOTC and IMO.It would be interesting if they take part in this discussion. – Siddharth Singh · 1 year, 1 month ago

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– Raushan Sharma · 1 year ago

Number Theory by Burton is the best for preparing for RMO's number theory part, as I think.Log in to reply

Excellent job! Keep on posting RMO type problems!

@Shivam Jadhav – Swapnil Das · 1 year, 1 month ago

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@Shivam Jadhav I appreciate your efforts of posting RMO problems. It would have been great if you posted proof problems also as note. (Like Xuming does for geometry). Thanks anyways for your step :) – Nihar Mahajan · 1 year, 1 month ago

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– Shivam Jadhav · 1 year, 1 month ago

Surely NiharLog in to reply

– Naitik Sanghavi · 1 year, 1 month ago

Hii I am also preparing for RMO can you tell me topics or chapters(syllabus) which we have to prepare for RMO...Log in to reply

– Raushan Sharma · 1 year ago

The major chapters are Number Theory, Algebra, Geometry and CombinatoricsLog in to reply

– Swapnil Das · 1 year, 1 month ago

Yes, I will provide you the complete Syllabus in few hours☺Log in to reply

– Saarthak Marathe · 1 year, 1 month ago

There's no as such particular syllabus for RMO.Log in to reply

Sir which book would u suggest for number theory – Abhishekrocks Sahoo · 1 year ago

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– Swapnil Das · 1 year ago

Hello sir, I have been introduced to David Burton's Number theory, which I have started using. I would recommend you the same.Log in to reply

– Abhishekrocks Sahoo · 1 year ago

Sir firstly i would like to appreciate this step of yours of making rmo board thank you :) and also thank you for your suggestionLog in to reply

– Swapnil Das · 1 year ago

Welcome , it was my pleasure benefiting you :)Log in to reply

@Swapnil Das Don't you think it would be better if you create a second part of this note? That way it will be better for people to see comments more easily and respond to them since there are so many comments in this note already. – Svatejas Shivakumar · 1 year ago

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new thread.Thanks. – Nihar Mahajan · 1 year ago

I liked you suggestion and I have created aLog in to reply

– Svatejas Shivakumar · 1 year ago

Thank you so much for your efforts.Log in to reply

– Nihar Mahajan · 1 year ago

No need to thank me. This is our continued effort :)Log in to reply

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– Nihar Mahajan · 1 year ago

No I don't think this will do good.Log in to reply

– Svatejas Shivakumar · 1 year ago

okLog in to reply

If \(p\) is a prime number, then prove that

is not a perfect square. – Dev Sharma · 1 year, 1 month ago\(7p + 3p -4\)Log in to reply

A]\(p\equiv 1 (mod 4) \): \(10p-4\equiv6 \equiv 2 (mod 4) \)

B] \(p\equiv -1 (mod 4) \) : \(10p-4\equiv -14 \equiv 2 (mod 4) \)

So neither of the two give us \( \equiv 0,1 (mod 4) \).

Therefore, there are no such square except for \( p=2 \) – Saarthak Marathe · 1 year, 1 month ago

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– Mehul Arora · 1 year, 1 month ago

Yep. This one is extremely easy.Log in to reply

It can be proved easily for \(p=2\) .

All perfect squares are \( \equiv 1,0 (mod 4) \)

We know that for all \(p\) excluding \(2\), \( {p}^{2}\equiv 1 (mod 4) \)

As all primes are odd numbers, we can segregate the primes into two cases:

A] \( p\equiv 1 (mod 4) \) :

For this, \( 7{p}^{2}+3p-4\equiv 7*1+3*1-4\equiv 6\equiv 2 (mod 4) \) Therefore this case has no squares formed.

B] \( p\equiv -1 (mod 4) \):

For this, \( 7{p}^{2}+3p-4\equiv 7*1+3*(-1)-4\equiv 0 (mod 4) \)

This case seems to satisfy the required condition.

For this we need to apply \( (mod 11) \). All squares are \( \equiv 0,1,3,4,5,9 (mod 11) \). This can be proved.

So we just need to check that \( 1,3,4,5,9 (mod 11) \) is not satisfied for any \( p\equiv 1,3,5,7,9 (mod 11) \) in the equation \(7{p}^{2}+3p-4 \).

For squares \( \equiv 0 (mod 11) \), We just need to check for \(p=11\) and we will find out that this neither gives us a square.

Therefore there are no squares of the form \(7{p}^{2}+3p-4\) – Saarthak Marathe · 1 year, 1 month ago

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– Saarthak Marathe · 1 year, 1 month ago

Is it \(7{p}^{2}+3p-4\)?Log in to reply

– Kushagra Sahni · 1 year, 1 month ago

Must be.Log in to reply

try this...

Let \(a\) be positive real number such that \(a^3 = 6(a + 1)\) then prove that

has no real roots. – Dev Sharma · 1 year, 1 month ago\(x^2 + ax + a^2 - 6 = 0\)Log in to reply

– Saarthak Marathe · 1 year, 1 month ago

doneLog in to reply

– Dev Sharma · 1 year, 1 month ago

showLog in to reply

Let \( a=b+2/b \)

Therefore, \( { \left( b+2/b \right) }^{ 3 }-6(b+2/b)-6=0 \)

Simplifying we get that, \( {b}^{6}-6{b}^{3}+8=0 \)

Therefore, \( {b}^3=4\) or \(2\)

Substitute these values to get \(a\).

That time we see that only one real solution of \(a\) occurs which is, \(a={2}^{1/3}+{2}^{2/3} \)

We see that, \( {a}^{2}-6=6/a\)

Substituting this value in \({x}^{2}+ax+{a}^{2}-6=0 \) we get that,

\(a{x}^{2}+{a}^{2}x+6=0 \)

Assume that the roots of these quadratic equation are real, Then using formula for roots for quadratic equations,

\( x=\frac { -a\pm \sqrt { { a }^{ 2 }-24 } }{ 2 } \)

Then substituting the acquired value of \(a\) in this equation we get that \(x\) is a complex number. Hence, our assumption was wrong.

Hence proved that roots of the given quadratic equations are not real. – Saarthak Marathe · 1 year, 1 month ago

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– Raushan Sharma · 6 months, 2 weeks ago

This can also be done using Cardano's method of finding solutions of a cubic equation.Log in to reply

Thanks in advance! :-)

Ingenious solution nonetheless! – Aniruddha Bhattacharjee · 6 months, 2 weeks ago

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– Mehul Arora · 1 year, 1 month ago

You're a genius. _/_Log in to reply

– Saarthak Marathe · 1 year, 1 month ago

Nah. Just able to solve RMO problemsLog in to reply

– Mehul Arora · 1 year, 1 month ago

Solving RMO probs is not easy bro ;)Log in to reply

– Saarthak Marathe · 1 year, 1 month ago

Probably yesLog in to reply

– Dev Sharma · 1 year, 1 month ago

nice... Try my another question which i am going to postLog in to reply

– Swapnil Das · 1 year, 1 month ago

Correct!Log in to reply

– Saarthak Marathe · 1 year, 1 month ago

Thanks!Log in to reply

Can someone suggest a good book for combinatorics with lots of examples and problems with solutions for RMO – Racchit Jain · 10 months, 1 week ago

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Guys , looking for varied solutions here.

Instead of posting questions here , we will post them as a note and give their respective links here. Is this okay? – Nihar Mahajan · 1 year, 1 month ago

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– Dev Sharma · 1 year, 1 month ago

ok!!Log in to reply

Do you have to be in Romania in order to qualify for RMO? – Alan Yan · 1 year, 1 month ago

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– Swapnil Das · 1 year, 1 month ago

No, it is the board of the Indian RMO.Log in to reply

– Alan Yan · 1 year, 1 month ago

Oh sorry, wrong RMO.Log in to reply

Has anybody heard of CHINESE DUMBASS NOTATION. (LOL) But keeping the name aside, its a very good tool for solving most of the types of inequalities in RMO. Read about this!! It is very helpful. – Saarthak Marathe · 1 year, 1 month ago

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– Nihar Mahajan · 1 year, 1 month ago

Yes , I have heard about it. But I have never applied it though :PLog in to reply

– Saarthak Marathe · 1 year, 1 month ago

Ok XDLog in to reply

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– Shrihari B · 10 months, 1 week ago

Wow Saarthak I didn't know you qualified RMO last year. Great achievement. I thought that u too like me had just cleared pre -rmoLog in to reply

Today, i got a call from rmo office, and they were saying that from my region only 3 student filled the form. They cant conduct exam on the preferred center by me. And they were saying i had to come to jaipur(capital)... -_- – Dev Sharma · 11 months, 3 weeks ago

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– Nihar Mahajan · 11 months, 3 weeks ago

Ohh , In which city and state do you live?Log in to reply

– Dev Sharma · 11 months, 3 weeks ago

I live in Nohar (northern rajasthan)..Log in to reply

– Satyajit Ghosh · 11 months, 3 weeks ago

Lol....It looks like you are already selected.Log in to reply

– Nihar Mahajan · 11 months, 3 weeks ago

How can a person get selected without giving the exam? Weird...Log in to reply

Ps-it was a joke and I didn't mean he is already selected. – Satyajit Ghosh · 11 months, 3 weeks ago

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– Dev Sharma · 11 months, 3 weeks ago

how?Log in to reply

@Calvin Lin Sir, is it possible for you to close this note since we already have a part 2 for this note. – Svatejas Shivakumar · 1 year ago

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– Calvin Lin Staff · 1 year ago

I don't see why this note should be locked.Log in to reply

– Swapnil Das · 1 year ago

Please sir, don't lock this note. I'm still benefiting from it.Log in to reply

@Swapnil Das @Mehul Arora @Dev Sharma Check out this link http://artofproblemsolving.com/community/c3176

indiacontests.It contains the problems of several contests held in India (including RMO,INMO and problems for the IMOTC). – Svatejas Shivakumar · 1 year agoLog in to reply

@Svatejas Shivakumar – Mehul Arora · 1 year ago

Thanks!Log in to reply

Try this problem https://brilliant.org/discussions/thread/rmo-practice-problems-from-past-year-papers/ – Svatejas Shivakumar · 1 year, 1 month ago

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RMO Set – Mehul Arora · 1 year, 1 month ago

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CHECK THIS OUT INEQUALITY. – Shivam Jadhav · 1 year, 1 month ago

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Hi,

So what is the

Theorem of the day?Any new topic?

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– Kritarth Lohomi · 1 year, 1 month ago

Chinese Remainder Theorem!!! ,would be the bestLog in to reply

Euler's Theorem– Swapnil Das · 1 year, 1 month agoLog in to reply

– Saarthak Marathe · 1 year, 1 month ago

Cauchy-Schwartz InequalityLog in to reply

can students of class xii participate in RMO? – Neeraj Snappy · 1 year, 1 month ago

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– Swapnil Das · 1 year, 1 month ago

Not really.Log in to reply

– Neeraj Snappy · 1 year, 1 month ago

are you sure, sir?Log in to reply

– Swapnil Das · 1 year, 1 month ago

Yes, Sir. 12th grade is now restricted to appear RMO.Log in to reply

Guys, if you want to solve a RMO problem, see this one https://brilliant.org/problems/a-geometry-problem-by-saarthak-marathe-2/?group=Z7UjgQAVmgvN . For more,see my sets. – Saarthak Marathe · 1 year, 1 month ago

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I have posted the note – Shivam Jadhav · 1 year, 1 month ago

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I have a doubt:

Find the sum of the squares of the roots of the equation :

\({ x }^{ 2 }+7[x]+5=0\) – Swapnil Das · 1 year, 1 month ago

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– Kushagra Sahni · 1 year, 1 month ago

He is wrong, they aren't integer solutions and he assumed in the beginning that x is an integer. Morover, it should have been 39.Log in to reply

– Kushagra Sahni · 1 year, 1 month ago

It should be 39 is case 1. And the values you are getting are not integers so do you think those solutions are valid?Log in to reply

– Kushagra Sahni · 1 year, 1 month ago

Very close to the answer, I will tell you today.Log in to reply

– Kushagra Sahni · 1 year, 1 month ago

Is this mod x? If it is then there are no solutions to this equation.Log in to reply

– Swapnil Das · 1 year, 1 month ago

No, it is the ceiling function.Log in to reply

– Kushagra Sahni · 1 year, 1 month ago

Why did you delete my comment? If it is the ceiling function then the answer is 92 and if it is the floor function then the answer is 95.Log in to reply

– Kushagra Sahni · 1 year, 1 month ago

Really, that means it is the smallest integer function, this question becomes easierLog in to reply

– Aditya Chauhan · 1 year, 1 month ago

It is floor functionLog in to reply

\( [-4.243252]=-5] \) – Saarthak Marathe · 1 year, 1 month ago

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– Kushagra Sahni · 1 year, 1 month ago

I know what is greatest integer function, but its notation is |_| is like this as it is the floor function.Log in to reply

– Saarthak Marathe · 1 year, 1 month ago

greatest integer function can be called as a floor function.Log in to reply

– Kushagra Sahni · 1 year, 1 month ago

That's what I said didn't I. I said it has the same notation because it is the floor function.Log in to reply

– Yuki Kuriyama · 1 year, 1 month ago

Is the answer -7 (by any chance) ?Log in to reply

– Kushagra Sahni · 1 year, 1 month ago

How can sum of squares be negative?Log in to reply

– Yuki Kuriyama · 1 year, 1 month ago

Oh..well--I think I overlooked the word "squares"..x'tremely sorry!!Log in to reply

OK, so the topic of the Day, from my side, is :

\(\huge\ Vieta's Formula\) – Swapnil Das · 1 year, 1 month ago

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– Satyajit Ghosh · 1 year, 1 month ago

Can you give the links from where you found them?Log in to reply

– Swapnil Das · 1 year, 1 month ago

The question?Log in to reply

– Satyajit Ghosh · 1 year, 1 month ago

Yeah the question or if you can find wiki's. Because there are many names of theorem which I don't know but wjen I see them, they are actually quite often used by meLog in to reply

– Dev Sharma · 1 year, 1 month ago

inequalitiesLog in to reply

– Swapnil Das · 1 year, 1 month ago

OK,good idea! Even I haven't started that topic😛Log in to reply

Can anyone share Topic of the day, so that we get to study it, and do some problems on it? – Swapnil Das · 1 year, 1 month ago

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– Shivam Jadhav · 1 year, 1 month ago

We must start with inequalitiesLog in to reply

Please someone tell me good brilliant questions that are good for RMO preparation except Shivam Jadhav's problems. – Kushagra Sahni · 1 year, 1 month ago

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– Swapnil Das · 1 year, 1 month ago

Try the set, " Openly welcome for future Mathematicians".Log in to reply

what about RMO forms? – Dev Sharma · 1 year, 1 month ago

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– Svatejas Shivakumar · 1 year, 1 month ago

The forms for some of the regions have already been uploaded. In which region are you giving RMO?Log in to reply

– Swapnil Das · 1 year, 1 month ago

Can you elaborate?Log in to reply

– Dev Sharma · 1 year, 1 month ago

i am asking about the date when forms would be availableLog in to reply

– Dev Sharma · 1 year, 1 month ago

do you know?????Log in to reply

http://olympiads.hbcse.tifr.res.in/enrollment/list-of-rmo-coordinators.Note that a region may be further divided into sub regions. You may see the website for your region or contact your regional coordinator for more details. – Svatejas Shivakumar · 1 year, 1 month ago

You can write in any of the regions(as per your convenience). See this link for the list of regions.Log in to reply

– Dev Sharma · 1 year, 1 month ago

i live in hanumangarh district in rajasthanLog in to reply

– Svatejas Shivakumar · 1 year, 1 month ago

I don't know about Rajasthan region much.You can contact your regional coordinator from the link above.Log in to reply

– Swapnil Das · 1 year, 1 month ago

Here, Pre RMO is kinda integer type exam. No proving😜Log in to reply

– Svatejas Shivakumar · 1 year, 1 month ago

No pre RMO in my region 😟Log in to reply

@Swapnil Das don't you think rmo is more of higher thinking with concept. Only concept is not what all it requires. – Satyajit Ghosh · 1 year, 1 month ago

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– Swapnil Das · 1 year, 1 month ago

Think of finding the Area of triangle without knowing the formula. Concept is the very fist thing to be cleared. After knowing varied concepts, brain works better and you can think stuff in a number of ways and directions.Log in to reply

– Nihar Mahajan · 1 year, 1 month ago

Yes , It requires out of box thinking too...Log in to reply

OMGOMGOMGOMGOMGOMGOMGOMGOMG!!!!!!! – Jingyang Tan · 1 year, 1 month ago

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– Swapnil Das · 1 year, 1 month ago

Delete the comment.Log in to reply

– Nihar Mahajan · 1 year, 1 month ago

Don't worry , the more he comments , the sooner his account will be deleted and he will be banned :)Log in to reply