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

Previous years questions
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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2^{34}

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\sqrt{2}

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Easy Math Editor

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:

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Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
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Note: you must add a full line of space before and after lists for them to show up correctly
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Math	Appears 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}`	$\boxed{123}$

Comments

Excellent job! Keep on posting RMO type problems!

@Shivam Jadhav

Swapnil Das - 5 years, 9 months ago

@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 - 5 years, 9 months ago

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

naitik sanghavi - 5 years, 9 months ago

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

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

Saarthak Marathe - 5 years, 9 months ago

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

Raushan Sharma - 5 years, 8 months ago

Surely Nihar

Shivam Jadhav - 5 years, 9 months ago

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 - 5 years, 9 months ago

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

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 - 5 years, 9 months ago

done

show

@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 – Correct!

@Swapnil Das – Thanks!

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

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

Mehul Arora - 5 years, 9 months ago

@Mehul Arora – Nah. Just able to solve RMO problems

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

@Mehul Arora – Probably yes

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

Thanks in advance! :-)

Ingenious solution nonetheless!

Aniruddha Bhattacharjee - 5 years, 2 months ago

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

Raushan Sharma - 5 years, 2 months ago

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

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 $p=2$

Yep. This one is extremely easy.

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 $7{p}^{2}+3p-4$

Is it $7{p}^{2}+3p-4$ ?

Must be.

Kushagra Sahni - 5 years, 9 months ago

@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.

A Former Brilliant Member - 5 years, 8 months ago

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

Nihar Mahajan - 5 years, 8 months ago

Thank you so much for your efforts.

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

Sir which book would u suggest for number theory

abhishekrocks sahoo - 5 years, 8 months ago

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 - 5 years, 8 months ago

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 – Welcome , it was my pleasure benefiting you :)

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.

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

Ok XD

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

Alan Yan - 5 years, 9 months ago

No, it is the board of the Indian RMO.

Oh sorry, wrong RMO.

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?

ok!!

Dev Sharma - 5 years, 8 months ago

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

Racchit Jain - 5 years, 6 months ago

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

Satyajit Ghosh - 5 years, 9 months 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.

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

what about RMO forms?

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

A Former Brilliant Member - 5 years, 9 months ago

Can you elaborate?

i am asking about the date when forms would be available

do you know?????

@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.

@A Former Brilliant Member – i live in hanumangarh district in rajasthan

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

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

@Swapnil Das – No pre RMO in my region 😟

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

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

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

We must start with inequalities

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

$\huge\ Vieta's Formula$

inequalities

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

Can you give the links from where you found them?

The question?

@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

I have a doubt:

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

${ x }^{ 2 }+7[x]+5=0$

Is the answer -7 (by any chance) ?

Yuki Kuriyama - 5 years, 9 months ago

How can sum of squares be negative?

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

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

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

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

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

No, it is the ceiling function.

@Swapnil Das – It is floor function

Aditya Chauhan - 5 years, 9 months ago

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

@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.

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

I have posted the note

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.

can students of class xii participate in RMO?

Neeraj Snappy - 5 years, 9 months ago

Not really.

are you sure, sir?

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

Hi,

So what is the Theorem of the day?
Any new topic?

Chinese Remainder Theorem!!! ,would be the best

kritarth lohomi - 5 years, 9 months ago

Cauchy-Schwartz Inequality

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

Euler's Theorem

CHECK THIS OUT INEQUALITY.

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

@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).

Thanks! @Svatejas Shivakumar

Mehul Arora - 5 years, 8 months ago

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

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

Calvin Lin Staff - 5 years, 8 months ago

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

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 - 5 years, 7 months ago

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

Satyajit Ghosh - 5 years, 7 months ago

how?

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

Nihar Mahajan - 5 years, 7 months ago

@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.

Ohh , In which city and state do you live?

I live in Nohar (northern rajasthan)..

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

Vishwash Kumar ΓΞΩ - 4 years, 2 months ago

OMGOMGOMGOMGOMGOMGOMGOMGOMG!!!!!!!

Jingyang Tan - 5 years, 9 months ago

Delete the comment.

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

Math	Appears 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}`	$\boxed{123}$