\(\color{Red}{\text{This year's (2013) RMO paper of Maharashtra and Goa Region .}}\)

(1) Find all positive integers \(m\) such that \((m-1)!\) is divisible by \(m\).

(2) \(H\) is the orthocentre of \(\triangle ABC\), \(D\) is any point on \(BC\) if a circle described with centre \(D\) and radius \(DH\) meets \(AH\) produced in \(E\), Prove that \(E\) lies on the circumcircle of \(\triangle ABC\).

(3) Suppose \(\triangle ABC\) is an acute angled triangle with \(AB<AC\). Let \(M\) be the midpoint of \(BC\). Suppose \(P\) is a point on side \(AB\) such that, if \(PC\) intersects the median \(AM\) at \(E\), then \(AP=PE\).Prove \(AB=CE\).

(4) Let \(x\) and y be real numbers such that \(x^2 + y^2 = 1\)

Prove that \(\dfrac {1}{x^2+1} +\dfrac {1}{y^2+1}+\dfrac {1}{1+xy} \geq \dfrac {3} {1+\frac {(x+y)^2}{4}}\)

(5) Let \(a_n\) be number of sequences of n terms formed using the digits \(0,1,2\) and \(3\) in which \(0\) occurs an odd number of times. Find \(a_n\)

(6) Find all positive integers \(n\) such that the product of all the positive divisors of \(n\) is equal to \(n^3\).

## Comments

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TopNewestThe third question is very simple one. One needs to use the result that when two cevians intersect the sides of triangles in the ratio m and n respectively, then they divide each other in the ratio m(n+1)/n and n(m+1)/m. Then the question becomes very simple. – Dinesh Chavan · 3 years, 8 months ago

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– Aditya Raut · 3 years, 8 months ago

Yes and that formula can be proved simply by Using B.P.T. or Basic Proportionality Theorem...Log in to reply

@Aditya Raut – Dev Sharma · 1 year, 11 months ago

which books did you use for RMO?Log in to reply

– Hrushikesh Bhope · 3 years, 8 months ago

I solved this one using B.P.T and similar triangles.Log in to reply

Find all positive integers n such that the product of all the positive divisors of n is equal to n^3 – Arun Patil · 3 years, 8 months ago

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– Aditya Raut · 3 years, 8 months ago

Simply find "n" who have 6 positive divisors(including n itself) thus giving 3 pairs that give product exactly \(n\) so that product of all divisors being \(n^3\)........ hence \(n\)=\(p^2_1.p_2\) OR n=\(p^5\) OR 1Log in to reply

@Aditya Raut – Anuj Shikarkhane · 2 years, 7 months ago

How come you have put more than 5 tags?Log in to reply

– Aditya Raut · 2 years, 7 months ago

This discussion was posted before January 2014, so at that time you could add as many tags as you want...Log in to reply

@Aditya Raut – Anuj Shikarkhane · 2 years, 7 months ago

Oh, btw congrats for clearing RMO.Log in to reply

Answer to 1st question:-

\(m \in \mathbb{N}- \) {4,p}

where \(p\) is prime. (All prime numbers are discarded) – Aditya Raut · 3 years, 8 months ago

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– Dinesh Chavan · 3 years, 8 months ago

To cancel all the primes, We can use Wilson's Theorem.Log in to reply

@Aditya Raut How to solve the 4 th question can you tell – Neel Khare · 5 months, 4 weeks ago

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Is my answer to question 5 correct? \(a_n = \frac{3^{n+1}-3}{8}\) if \(n\) is even and \(a_n = \frac{3^n-1}{8}\) if \(n\) is odd?

By the way, I have a few more questions. Was this the same paper for Pune region? Is Arthur Engel's book good? – Arulx Z · 1 year ago

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how did we solve question number 4 – Rahul Rawat · 1 year, 7 months ago

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What is the answer to question 5? – Shashank Rammoorthy · 1 year, 8 months ago

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please give me the proof of weierstrauss product inequality – Dipak Bhole · 2 years, 8 months ago

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Here is your link :) – Aditya Raut · 2 years, 8 months ago

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– Dipak Bhole · 2 years, 3 months ago

what is the jensens inequality?Log in to reply

Why's this incomplete? @Aditya Raut -Care to complete it? – Krishna Ar · 3 years ago

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– Aditya Raut · 3 years ago

DONE .... \(\color{Red}{T} \color{Orange}{H} \color{LimeGreen}{A} \color{Green}{N} \color{Blue}{K} \color{Purple}{S} \color{Pink}{S} \color{DarkRed}{S}\) .... If you hadn't commented, i wouldn't have known that this paper was later made into half paper.... that might be a mistake by some staff most probably...Log in to reply

@Aditya Raut -I guess I'd be appearing for RMO dis'yr (That depends on my pre-RMO performance though :P). I;d like to have some tips and prep-method from you. I know that the MG-RMO is much different and easy compared to TN-RMO( which is why we have an equally tough Pre-RMO :P ). ..but yet...You're too good in math...so....even book references would be much appreciated. And-another thing- could you tell me a good resource to learn parametric substitution to solve algebraic equations? I recently noticed this in some problem on this site- and found it fun...I want to know more...Thanks :) – Krishna Ar · 3 years ago

Hi Again!Log in to reply

If you don't mid give me your email ID plz

And when you reply to this, please mention me. (Hope you'll reply to say thanks :P LOL)

For RMO tips, I recommend you to do what Dinesh said in the discussion of this problem , because me and Dinesh are classmates and I have already said that do what he said (though he didn't clear RMO with me last year, he has double potential to do that).

Book references that I have been using are \(\color{Red}{\textbf{Inequalities- An Approach Through Problems}}\) (author B. J. Venkatachala),

Inequalities

\(\color{Red}{\textbf{Applied Combinatorics}}\) (author Alan Tucker)

App. Comb

\(\color{Red}{\textbf{Elementary Number Theory}}\) (author David M. Burton)

nt

\(\color{Red}{\textbf{Problem Solving Strategies}}\) (author Arthur Engel)

PSS

\(\color{Red}{\textbf{An Excursion in Mathematics}}\) (by Bhaskaracharya Pratishthana, Pune)

\(\color{Green}{\text{This one i got from Bhaskaracharya Pratishthana for clearing RMO}}\) \(\color{Green}{\text{(The co-oridinator of RMO in Maharashtra and Goa region)}}\)

\(\color{Red}{\textbf{IMO problems}}\) (author Istvan Reiman)

1

2

3

Try previous year RMO papers, that is the most useful thing. Don't look at solutions till you have tried it a lot. And most important, keep calm, because in RMO, every question is solvable, but the \(\color{Blue}{\textbf{TRICK}}\) doesn't \(\color{Blue}{\textbf{CLICK}}\) at the time, so think on small things.

That's all i can tell, if you want to know more books, or have all the above books for free

(To get these books, I paid ₹ \(3379/-\) , but you can get them for free)

Just go to libgen.org or go to bookzz.org and search the book, i am sure you'll get for free. (I downloaded a very costly book named "Learn Python the hard way" for free from here).

@Krishna Ar , @Dinesh Chavan – Aditya Raut · 3 years ago

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– Akash Deep · 3 years ago

bro thanks for giving such a useful websiteLog in to reply

img

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@Aditya Raut . I shall try to see the parametric stuff related things on net. Thx again! :) Shall I send you a mail? I guess I saw it in the sol of a problem? – Krishna Ar · 3 years ago

Wow...Thanks A lot for replyingLog in to reply

@Satvik Golechha too, please note this. – Aditya Raut · 3 years ago

Anytime, I am there for anyone who wants to be friends! By the way, please send me email if you want, at \(\color{Blue}{\textbf{adityaraut34@gmail.com}}\) instead of 'adityaraut34@yahoo.in', because my yahoo account is related to brilliant and so it is stuffed with emails of "someone liked/reshared/followed" , so for our communication, i prefer GMAIL. Same Applies to my new friendLog in to reply

– Krishna Ar · 3 years ago

Sure..thanks for notifying this too :P :DLog in to reply

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– Krishna Ar · 3 years ago

Oh..I am so sorry..that I didnt turn up..I'll send you a mail in 2 mins..asap :PLog in to reply

– Krishna Ar · 3 years ago

Thanks a ton!!!!! :)Log in to reply

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– Jayakumar Krishnan · 3 years ago

But...its not there...can you please complete it ?Log in to reply

I'm selected for the second level. Are you ??? – Hrishikesh Dani · 3 years, 7 months ago

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question no. 5 is also very easy. – Akhilesh Agrawal · 3 years, 8 months ago

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i will try – Sanskruti Lohia · 3 years, 8 months ago

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Answer to the 1st one : (m - 1)! is divisible for all positive integers except 4 and primes.

Answer to the 6th one : If you consider n and 1 to be the divisors too, the answers are p^5 and p^2 * q , p and q are primes.

Answer to the 4th one: Simplify the L.H.S and R.H.S of the inequality, such that both the sides contain only xy. After that, using the fact that ( x -y )^2 is always greater than or equal to zero, find out the max value of xy. Plug the max value of xy in the inequality. – Hrushikesh Bhope · 3 years, 8 months ago

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– Aditya Raut · 3 years, 8 months ago

6th problem also has 1as it's solution because positive divisor of 1 is 1 which satisfies product of divisors = \(1^3\)Log in to reply

– Aditya Raut · 3 years, 8 months ago

Yeah....we got \(xy \leq \frac {1}{2} \) by A.M- G.M inequality to numbers \(x^2 , y^2\)Log in to reply

– Ranjana Kasangeri · 3 years, 8 months ago

I used Titu's LemmaLog in to reply

– Dinesh Chavan · 3 years, 8 months ago

I used Cauchy's inequality to get the results.Log in to reply

Answer to 6th question:-

All n such that n=\(p^2_1\times p_2\) OR n=\(p^5\) OR 1

where p denotes prime......... – Aditya Raut · 3 years, 8 months ago

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where can I find answer key – Gopalkrishna Nayak Pangal · 2 years, 8 months ago

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How to solve the 2nd question? – Aditya Patil · 3 years, 7 months ago

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The third one is actually quite simple.......... Construct a line parallel to BC from P. Equate the ratio of AP and AB to the ratio of PE and EC ( This can be done by using B.P.T and similar triangles. ) – Hrushikesh Bhope · 3 years, 8 months ago

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