RMO Paper 2013 of Maharashtra and Goa region

This year’s (2013) RMO paper of Maharashtra and Goa Region .\color{#D61F06}{\text{This year's (2013) RMO paper of Maharashtra and Goa Region .}}

(1) Find all positive integers mm such that (m1)!(m-1)! is divisible by mm.

(2) HH is the orthocentre of ABC\triangle ABC, DD is any point on BCBC if a circle described with centre DD and radius DHDH meets AHAH produced in EE, Prove that EE lies on the circumcircle of ABC\triangle ABC.

(3) Suppose ABC\triangle ABC is an acute angled triangle with AB<ACAB<AC. Let MM be the midpoint of BCBC. Suppose PP is a point on side ABAB such that, if PCPC intersects the median AMAM at EE, then AP=PEAP=PE.Prove AB=CEAB=CE.

(4) Let xx and y be real numbers such that x2+y2=1x^2 + y^2 = 1

Prove that 1x2+1+1y2+1+11+xy31+(x+y)24\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 ana_n be number of sequences of n terms formed using the digits 0,1,20,1,2 and 33 in which 00 occurs an odd number of times. Find ana_n

(6) Find all positive integers nn such that the product of all the positive divisors of nn is equal to n3n^3.

Note by Aditya Raut
5 years, 10 months ago

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7 votes

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Find all positive integers n such that the product of all the positive divisors of n is equal to n^3

arun patil - 5 years, 10 months ago

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Simply find "n" who have 6 positive divisors(including n itself) thus giving 3 pairs that give product exactly nn so that product of all divisors being n3n^3........ hence nn=p12.p2p^2_1.p_2 OR n=p5p^5 OR 1

Aditya Raut - 5 years, 10 months ago

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How come you have put more than 5 tags?@Aditya Raut

Anuj Shikarkhane - 4 years, 9 months ago

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@Anuj Shikarkhane This discussion was posted before January 2014, so at that time you could add as many tags as you want...

Aditya Raut - 4 years, 9 months ago

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@Aditya Raut Oh, btw congrats for clearing RMO. @Aditya Raut

Anuj Shikarkhane - 4 years, 9 months ago

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The 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 - 5 years, 10 months ago

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

Aditya Raut - 5 years, 10 months ago

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which books did you use for RMO? @Aditya Raut

Dev Sharma - 4 years, 1 month ago

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PlZ give this ques Sol. Actually I am not able to solve this

AVNISH GARG - 2 years ago

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Can someone tell me where I can read more about this, and what is the name of this theorem?

Ram Keswani - 2 months, 1 week ago

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I solved this one using B.P.T and similar triangles.

Hrushikesh bhope - 5 years, 10 months ago

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Answer to 1st question:-

mNm \in \mathbb{N}- {4,p}

where pp is prime. (All prime numbers are discarded)

Aditya Raut - 5 years, 10 months ago

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To cancel all the primes, We can use Wilson's Theorem.

Dinesh Chavan - 5 years, 10 months ago

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how did we solve question number 4

Rahul Rawat - 3 years, 9 months ago

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Answer to 6th question:-

All n such that n=p12×p2p^2_1\times p_2 OR n=p5p^5 OR 1

where p denotes prime.........

Aditya Raut - 5 years, 10 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 - 5 years, 10 months ago

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I used Cauchy's inequality to get the results.

Dinesh Chavan - 5 years, 10 months ago

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I used Titu's Lemma

Ranjana Kasangeri - 5 years, 10 months ago

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Yeah....we got xy12xy \leq \frac {1}{2} by A.M- G.M inequality to numbers x2,y2x^2 , y^2

Aditya Raut - 5 years, 10 months ago

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6th problem also has 1as it's solution because positive divisor of 1 is 1 which satisfies product of divisors = 131^3

Aditya Raut - 5 years, 10 months ago

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1 and 6 have answers there are infinite solutions

Anurag Ramachandran - 2 years, 1 month ago

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i will try

Sanskruti Lohia - 5 years, 10 months ago

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question no. 5 is also very easy.

akhilesh agrawal - 5 years, 10 months ago

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I'm selected for the second level. Are you ???

Hrishikesh Dani - 5 years, 9 months ago

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Why's this incomplete? @Aditya Raut -Care to complete it?

Krishna Ar - 5 years, 2 months ago

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DONE .... THANKSSS\color{#D61F06}{T} \color{#EC7300}{H} \color{limegreen}{A} \color{#20A900}{N} \color{#3D99F6}{K} \color{#69047E}{S} \color{#E81990}{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...

Aditya Raut - 5 years, 2 months ago

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Thanks a ton!!!!! :)

Krishna Ar - 5 years, 2 months ago

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Hi Again! @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 - 5 years, 2 months ago

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@Krishna Ar Very good, about your learning parametric doubt first, you can try it out on the practice section of brilliant! Or just google it and see all details on the wolfram alpha forum type of website.

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 Inequalities- An Approach Through Problems\color{#D61F06}{\textbf{Inequalities- An Approach Through Problems}} (author B. J. Venkatachala),

Inequalities Inequalities


Applied Combinatorics\color{#D61F06}{\textbf{Applied Combinatorics}} (author Alan Tucker)

App. Comb App. Comb


Elementary Number Theory\color{#D61F06}{\textbf{Elementary Number Theory}} (author David M. Burton)

nt nt


Problem Solving Strategies\color{#D61F06}{\textbf{Problem Solving Strategies}} (author Arthur Engel)

PSS PSS


An Excursion in Mathematics\color{#D61F06}{\textbf{An Excursion in Mathematics}} (by Bhaskaracharya Pratishthana, Pune)

This one i got from Bhaskaracharya Pratishthana for clearing RMO\color{#20A900}{\text{This one i got from Bhaskaracharya Pratishthana for clearing RMO}} (The co-oridinator of RMO in Maharashtra and Goa region)\color{#20A900}{\text{(The co-oridinator of RMO in Maharashtra and Goa region)}}


IMO problems\color{#D61F06}{\textbf{IMO problems}} (author Istvan Reiman)

1 1 2 2 3 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 TRICK\color{#3D99F6}{\textbf{TRICK}} doesn't CLICK\color{#3D99F6}{\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 ₹ 3379/3379/- , 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 - 5 years, 2 months ago

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@Aditya Raut bro thanks for giving such a useful website

akash deep - 5 years, 2 months ago

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@Akash Deep Anytime ! Wanna be friends ? Gmail- adityaraut34@gmail.com \color{#3D99F6}{\text{adityaraut34@gmail.com }}

img img

Aditya Raut - 5 years, 2 months ago

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@Aditya Raut Wow...Thanks A lot for replying @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 - 5 years, 2 months ago

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@Krishna Ar Anytime, I am there for anyone who wants to be friends! By the way, please send me email if you want, at adityaraut34@gmail.com\color{#3D99F6}{\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 friend @Satvik Golechha too, please note this.

Aditya Raut - 5 years, 2 months ago

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@Aditya Raut Sure..thanks for notifying this too :P :D

Krishna Ar - 5 years, 2 months ago

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please give me the proof of weierstrauss product inequality

dipak bhole - 4 years, 10 months ago

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Here is your link :)

Aditya Raut - 4 years, 10 months ago

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what is the jensens inequality?

dipak bhole - 4 years, 5 months ago

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What is the answer to question 5?

Shashank Rammoorthy - 3 years, 10 months ago

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Is my answer to question 5 correct? an=3n+138a_n = \frac{3^{n+1}-3}{8} if nn is even and an=3n18a_n = \frac{3^n-1}{8} if nn 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 - 3 years, 1 month ago

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@Aditya Raut How to solve the 4 th question can you tell

Neel Khare - 2 years, 7 months ago

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pls explain how to solve the 4th one

neelmadhav sahu - 2 years ago

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You can use cauchy and (x-y)^2 greater than or equal to 0 for 4th question

Kinshuk Bansal - 9 hours 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 - 5 years, 10 months ago

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How to solve the 2nd question?

Aditya Patil - 5 years, 9 months ago

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where can I find answer key

Gopalkrishna Nayak Pangal - 4 years, 10 months ago

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