# 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
4 years, 1 month ago

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

- 2 years, 6 months ago

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

- 3 years, 10 months ago

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)... -_-

- 3 years, 11 months ago

Ohh , In which city and state do you live?

- 3 years, 11 months ago

I live in Nohar (northern rajasthan)..

- 3 years, 11 months ago

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

- 3 years, 11 months ago

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

- 3 years, 11 months ago

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.

- 3 years, 11 months ago

how?

- 3 years, 11 months ago

Sir which book would u suggest for number theory

- 4 years ago

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

- 4 years 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

- 4 years ago

Welcome , it was my pleasure benefiting you :)

- 4 years 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.

Staff - 4 years ago

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

- 4 years 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.

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

- 4 years ago

Thank you so much for your efforts.

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

- 4 years ago

@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

- 4 years ago

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?

- 4 years ago

ok!!

- 4 years ago

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

- 4 years, 1 month ago

No, it is the board of the Indian RMO.

- 4 years, 1 month ago

Oh sorry, wrong RMO.

- 4 years, 1 month ago

CHECK THIS OUT INEQUALITY.

- 4 years, 1 month ago

Hi,

• So what is the Theorem of the day?

• Any new topic?

- 4 years, 1 month ago

Chinese Remainder Theorem!!! ,would be the best

- 4 years ago

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

Euler's Theorem

- 4 years, 1 month ago

Cauchy-Schwartz Inequality

- 4 years, 1 month ago

can students of class xii participate in RMO?

- 4 years, 1 month ago

Not really.

- 4 years, 1 month ago

are you sure, sir?

- 4 years, 1 month ago

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

- 4 years, 1 month ago

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

- 4 years, 1 month ago

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 $p=2$

- 4 years, 1 month ago

Yep. This one is extremely easy.

- 4 years, 1 month ago

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

- 4 years, 1 month ago

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

- 4 years, 1 month ago

Must be.

- 4 years, 1 month ago

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.

- 4 years, 1 month ago

done

- 4 years, 1 month ago

show

- 4 years, 1 month ago

${a}^{3}-6a-6=0$

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.

- 4 years, 1 month ago

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

- 3 years, 6 months ago

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

Ingenious solution nonetheless!

- 3 years, 6 months ago

You're a genius. _/_

- 4 years, 1 month ago

Nah. Just able to solve RMO problems

- 4 years, 1 month ago

Solving RMO probs is not easy bro ;)

- 4 years, 1 month ago

Probably yes

- 4 years, 1 month ago

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

- 4 years, 1 month ago

Correct!

- 4 years, 1 month ago

Thanks!

- 4 years, 1 month ago

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.

- 4 years, 1 month ago

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

- 4 years, 1 month ago

Ok XD

- 4 years, 1 month ago

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.

- 4 years, 1 month ago

I have posted the note

- 4 years, 1 month ago

I have a doubt:

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

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

- 4 years, 1 month ago

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

- 4 years, 1 month ago

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

- 4 years, 1 month ago

No, it is the ceiling function.

- 4 years, 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.

- 4 years, 1 month ago

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

- 4 years, 1 month ago

It is floor function

- 4 years, 1 month ago

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

$[-4.243252]=-5]$

- 4 years, 1 month ago

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

- 4 years, 1 month ago

greatest integer function can be called as a floor function.

- 4 years, 1 month ago

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

- 4 years, 1 month ago

Is the answer -7 (by any chance) ?

- 4 years, 1 month ago

How can sum of squares be negative?

- 4 years, 1 month ago

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

- 4 years, 1 month ago

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

$\huge\ Vieta's Formula$

- 4 years, 1 month ago

Can you give the links from where you found them?

- 4 years, 1 month ago

The question?

- 4 years, 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 me

- 4 years, 1 month ago

inequalities

- 4 years, 1 month ago

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

- 4 years, 1 month ago

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

- 4 years, 1 month ago

- 4 years, 1 month ago

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

- 4 years, 1 month ago

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

- 4 years, 1 month ago

- 4 years, 1 month ago

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

- 4 years, 1 month ago

Can you elaborate?

- 4 years, 1 month ago

do you know?????

- 4 years, 1 month ago

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.

- 4 years, 1 month ago

i live in hanumangarh district in rajasthan

- 4 years, 1 month ago

- 4 years, 1 month ago

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

- 4 years, 1 month ago

No pre RMO in my region 😟

- 4 years, 1 month ago

- 4 years, 1 month 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.

- 4 years, 1 month ago

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

- 4 years ago

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

- 4 years, 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.

- 4 years, 1 month ago

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

- 4 years, 1 month ago

OMGOMGOMGOMGOMGOMGOMGOMGOMG!!!!!!!

- 4 years, 1 month ago

Delete the comment.

- 4 years, 1 month ago

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

- 4 years, 1 month ago

Excellent job! Keep on posting RMO type problems!

- 4 years, 1 month 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 :)

- 4 years, 1 month ago

Surely Nihar

- 4 years, 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...

- 4 years, 1 month ago

The major chapters are Number Theory, Algebra, Geometry and Combinatorics

- 4 years ago

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

- 4 years, 1 month ago

There's no as such particular syllabus for RMO.

- 4 years, 1 month ago