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**!

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## Comments

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TopNewestExcellent job! Keep on posting RMO type problems!

@Shivam Jadhav

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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 :)

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

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Surely Nihar

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Great 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.

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

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try this...

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

has no real roots.\(x^2 + ax + a^2 - 6 = 0\)Log in to reply

done

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show

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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.

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Thanks in advance! :-)

Ingenious solution nonetheless!

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

is not a perfect square.\(7p + 3p -4\)Log in to reply

If it is \(7p+3p-4=10p-4\) then it is extremely trivial. Applying the same logic as in the answer below,

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 \( p=2 \)

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Yep. This one is extremely easy.

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If its \(7{p}^{2}+3p-4\) then,

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 \(7{p}^{2}+3p-4\)

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

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Must be.

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@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.

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I liked you suggestion and I have created a new thread.Thanks.

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Thank you so much for your efforts.

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Sir which book would u suggest for number theory

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

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

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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.

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Yes , I have heard about it. But I have never applied it though :P

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Ok XD

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Do you have to be in Romania in order to qualify for RMO?

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No, it is the board of the Indian RMO.

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Oh sorry, wrong RMO.

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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?

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ok!!

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Can someone suggest a good book for combinatorics with lots of examples and problems with solutions for RMO

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@Swapnil Das don't you think rmo is more of higher thinking with concept. Only concept is not what all it requires.

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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.

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Yes , It requires out of box thinking too...

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what about RMO forms?

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

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Can you elaborate?

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i am asking about the date when forms would be available

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do you know?????

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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.

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

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Please someone tell me good brilliant questions that are good for RMO preparation except Shivam Jadhav's problems.

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Try the set, " Openly welcome for future Mathematicians".

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Can anyone share Topic of the day, so that we get to study it, and do some problems on it?

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We must start with inequalities

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

\(\huge\ Vieta's Formula\)

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inequalities

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OK,good idea! Even I haven't started that topic😛

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Can you give the links from where you found them?

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The question?

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I have a doubt:

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

\({ x }^{ 2 }+7[x]+5=0\)

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Is the answer -7 (by any chance) ?

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How can sum of squares be negative?

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

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No. It is greatest integer function, which means greatest integer less than the given number. For example, \( [3.23423]=3 \)

\( [-4.243252]=-5] \)

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No, it is the ceiling function.

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

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I have posted the note

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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.

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can students of class xii participate in RMO?

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Not really.

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are you sure, sir?

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Hi,

So what is the

Theorem of the day?Any new topic?

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Chinese Remainder Theorem!!! ,would be the best

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Cauchy-Schwartz Inequality

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OK, the topic of the day from my side is:

Euler's TheoremLog in to reply

CHECK THIS OUT INEQUALITY.

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Try this problem https://brilliant.org/discussions/thread/rmo-practice-problems-from-past-year-papers/

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@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).Log in to reply

Thanks! @Svatejas Shivakumar

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@Calvin Lin Sir, is it possible for you to close this note since we already have a part 2 for this note.

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I don't see why this note should be locked.

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

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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)... -_-

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Lol....It looks like you are already selected.

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how?

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How can a person get selected without giving the exam? Weird...

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Ps-it was a joke and I didn't mean he is already selected.

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Ohh , In which city and state do you live?

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I live in Nohar (northern rajasthan)..

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Is it closed now? A second question : Is \(GEOMETRY\) banned here? Not a single stuff....

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OMGOMGOMGOMGOMGOMGOMGOMGOMG!!!!!!!

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Delete the comment.

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

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