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RMO board

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

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!

Note by Swapnil Das
2 years, 1 month ago

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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.

Siddharth Singh - 2 years, 1 month ago

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

Raushan Sharma - 2 years ago

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Excellent job! Keep on posting RMO type problems!

@Shivam Jadhav

Swapnil Das - 2 years, 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 - 2 years, 1 month ago

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Surely Nihar

Shivam Jadhav - 2 years, 1 month ago

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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...

Naitik Sanghavi - 2 years, 1 month ago

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@Naitik Sanghavi The major chapters are Number Theory, Algebra, Geometry and Combinatorics

Raushan Sharma - 2 years ago

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@Naitik Sanghavi Yes, I will provide you the complete Syllabus in few hours☺

Swapnil Das - 2 years, 1 month ago

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@Swapnil Das There's no as such particular syllabus for RMO.

Saarthak Marathe - 2 years, 1 month ago

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Sir which book would u suggest for number theory

Abhishekrocks Sahoo - 2 years ago

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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.

Swapnil Das - 2 years ago

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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

Abhishekrocks Sahoo - 2 years ago

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@Abhishekrocks Sahoo Welcome , it was my pleasure benefiting you :)

Swapnil Das - 2 years ago

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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.

Brilliant Member - 2 years ago

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

Nihar Mahajan - 2 years ago

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Thank you so much for your efforts.

Brilliant Member - 2 years ago

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@Brilliant Member No need to thank me. This is our continued effort :)

Nihar Mahajan - 2 years ago

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Comment deleted Sep 26, 2015

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@Brilliant Member No I don't think this will do good.

Nihar Mahajan - 2 years ago

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@Nihar Mahajan ok

Brilliant Member - 2 years ago

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If \(p\) is a prime number, then prove that \(7p + 3p -4\) is not a perfect square.

Dev Sharma - 2 years, 1 month ago

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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 \( p=2 \)

Saarthak Marathe - 2 years, 1 month ago

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Yep. This one is extremely easy.

Mehul Arora - 2 years, 1 month ago

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

Saarthak Marathe - 2 years, 1 month ago

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

Saarthak Marathe - 2 years, 1 month ago

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Must be.

Kushagra Sahni - 2 years, 1 month ago

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try this...

Let \(a\) be positive real number such that \(a^3 = 6(a + 1)\) then prove that \(x^2 + ax + a^2 - 6 = 0\) has no real roots.

Dev Sharma - 2 years, 1 month ago

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done

Saarthak Marathe - 2 years, 1 month ago

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show

Dev Sharma - 2 years, 1 month ago

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@Dev Sharma \({a}^{3}-6a-6=0\)

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 - 2 years, 1 month ago

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@Saarthak Marathe This can also be done using Cardano's method of finding solutions of a cubic equation.

Raushan Sharma - 1 year, 6 months ago

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@Saarthak Marathe Can you explain why you took the initial substitution of a= b+ (2/b) ?

Thanks in advance! :-)

Ingenious solution nonetheless!

Aniruddha Bhattacharjee - 1 year, 6 months ago

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@Saarthak Marathe You're a genius. _/_

Mehul Arora - 2 years, 1 month ago

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@Mehul Arora Nah. Just able to solve RMO problems

Saarthak Marathe - 2 years, 1 month ago

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@Saarthak Marathe Solving RMO probs is not easy bro ;)

Mehul Arora - 2 years, 1 month ago

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@Mehul Arora Probably yes

Saarthak Marathe - 2 years, 1 month ago

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@Saarthak Marathe nice... Try my another question which i am going to post

Dev Sharma - 2 years, 1 month ago

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@Saarthak Marathe Correct!

Swapnil Das - 2 years, 1 month ago

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@Swapnil Das Thanks!

Saarthak Marathe - 2 years, 1 month ago

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

Racchit Jain - 1 year, 10 months 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 - 2 years ago

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ok!!

Dev Sharma - 2 years ago

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

Alan Yan - 2 years, 1 month ago

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No, it is the board of the Indian RMO.

Swapnil Das - 2 years, 1 month ago

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Oh sorry, wrong RMO.

Alan Yan - 2 years, 1 month ago

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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.

Saarthak Marathe - 2 years, 1 month ago

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

Nihar Mahajan - 2 years, 1 month ago

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Ok XD

Saarthak Marathe - 2 years, 1 month ago

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Comment deleted Dec 18, 2015

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

Shrihari B - 1 year, 10 months ago

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

Rohit Camfar - 6 months, 3 weeks ago

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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)... -_-

Dev Sharma - 1 year, 11 months ago

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Ohh , In which city and state do you live?

Nihar Mahajan - 1 year, 11 months ago

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I live in Nohar (northern rajasthan)..

Dev Sharma - 1 year, 11 months ago

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

Satyajit Ghosh - 1 year, 11 months ago

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

Nihar Mahajan - 1 year, 11 months ago

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@Nihar Mahajan No I meant that in any region (at least for Delhi 17 are selected) at least 20 are selected. So even getting low marks you have a high probability for selection.

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

Satyajit Ghosh - 1 year, 11 months ago

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how?

Dev Sharma - 1 year, 11 months ago

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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.

Brilliant Member - 2 years ago

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

Calvin Lin Staff - 2 years ago

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

Swapnil Das - 2 years ago

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@Swapnil Das @Mehul Arora @Dev Sharma Check out this link http://artofproblemsolving.com/community/c3176indiacontests.It contains the problems of several contests held in India (including RMO,INMO and problems for the IMOTC).

Brilliant Member - 2 years ago

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Thanks! @Svatejas Shivakumar

Mehul Arora - 2 years ago

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Comment deleted Sep 17, 2015

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RMO Set

Mehul Arora - 2 years, 1 month ago

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CHECK THIS OUT INEQUALITY.

Shivam Jadhav - 2 years, 1 month ago

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Hi,

  • So what is the Theorem of the day?

  • Any new topic?

Swapnil Das - 2 years, 1 month ago

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Chinese Remainder Theorem!!! ,would be the best

Kritarth Lohomi - 2 years, 1 month ago

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

Euler's Theorem

Swapnil Das - 2 years, 1 month ago

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Cauchy-Schwartz Inequality

Saarthak Marathe - 2 years, 1 month ago

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can students of class xii participate in RMO?

Neeraj Snappy - 2 years, 1 month ago

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Not really.

Swapnil Das - 2 years, 1 month ago

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are you sure, sir?

Neeraj Snappy - 2 years, 1 month ago

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@Neeraj Snappy Yes, Sir. 12th grade is now restricted to appear RMO.

Swapnil Das - 2 years, 1 month ago

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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.

Saarthak Marathe - 2 years, 1 month ago

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I have posted the note

Shivam Jadhav - 2 years, 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 - 2 years, 1 month ago

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Comment deleted Sep 18, 2015

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Comment deleted Sep 17, 2015

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@Swapnil Das 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.

Kushagra Sahni - 2 years, 1 month ago

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

Kushagra Sahni - 2 years, 1 month ago

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

Kushagra Sahni - 2 years, 1 month ago

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

Kushagra Sahni - 2 years, 1 month ago

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No, it is the ceiling function.

Swapnil Das - 2 years, 1 month ago

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@Swapnil Das 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.

Kushagra Sahni - 2 years, 1 month ago

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@Swapnil Das Really, that means it is the smallest integer function, this question becomes easier

Kushagra Sahni - 2 years, 1 month ago

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@Swapnil Das It is floor function

Aditya Chauhan - 2 years, 1 month ago

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

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

Saarthak Marathe - 2 years, 1 month ago

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@Saarthak Marathe I know what is greatest integer function, but its notation is |_| is like this as it is the floor function.

Kushagra Sahni - 2 years, 1 month ago

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@Kushagra Sahni greatest integer function can be called as a floor function.

Saarthak Marathe - 2 years, 1 month ago

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@Saarthak Marathe That's what I said didn't I. I said it has the same notation because it is the floor function.

Kushagra Sahni - 2 years, 1 month ago

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Is the answer -7 (by any chance) ?

Yuki Kuriyama - 2 years, 1 month ago

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How can sum of squares be negative?

Kushagra Sahni - 2 years, 1 month ago

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@Kushagra Sahni Oh..well--I think I overlooked the word "squares"..x'tremely sorry!!

Yuki Kuriyama - 2 years, 1 month ago

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

\(\huge\ Vieta's Formula\)

Swapnil Das - 2 years, 1 month ago

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Can you give the links from where you found them?

Satyajit Ghosh - 2 years, 1 month ago

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The question?

Swapnil Das - 2 years, 1 month ago

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@Swapnil Das 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 me

Satyajit Ghosh - 2 years, 1 month ago

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inequalities

Dev Sharma - 2 years, 1 month ago

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

Swapnil Das - 2 years, 1 month ago

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

Swapnil Das - 2 years, 1 month ago

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We must start with inequalities

Shivam Jadhav - 2 years, 1 month ago

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

Kushagra Sahni - 2 years, 1 month ago

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

Swapnil Das - 2 years, 1 month ago

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what about RMO forms?

Dev Sharma - 2 years, 1 month ago

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

Brilliant Member - 2 years, 1 month ago

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Can you elaborate?

Swapnil Das - 2 years, 1 month ago

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i am asking about the date when forms would be available

Dev Sharma - 2 years, 1 month ago

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do you know?????

Dev Sharma - 2 years, 1 month ago

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@Dev Sharma You can write in any of the regions(as per your convenience). See this link for the list of regions.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.

Brilliant Member - 2 years, 1 month ago

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@Brilliant Member i live in hanumangarh district in rajasthan

Dev Sharma - 2 years, 1 month ago

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@Dev Sharma I don't know about Rajasthan region much.You can contact your regional coordinator from the link above.

Brilliant Member - 2 years, 1 month ago

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@Brilliant Member Here, Pre RMO is kinda integer type exam. No proving😜

Swapnil Das - 2 years, 1 month ago

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@Swapnil Das No pre RMO in my region 😟

Brilliant Member - 2 years, 1 month ago

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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.

Satyajit Ghosh - 2 years, 1 month ago

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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.

Swapnil Das - 2 years, 1 month ago

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

Nihar Mahajan - 2 years, 1 month ago

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OMGOMGOMGOMGOMGOMGOMGOMGOMG!!!!!!!

Jingyang Tan - 2 years, 1 month ago

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Delete the comment.

Swapnil Das - 2 years, 1 month ago

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

Nihar Mahajan - 2 years, 1 month ago

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