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

(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 \(n^3\).

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## Comments

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

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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 1

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

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@Aditya Raut

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

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.

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

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which books did you use for RMO? @Aditya Raut

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

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

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Answer to 1st question:-

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

where \(p\) is prime. (All prime numbers are discarded)

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

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how did we solve question number 4

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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.........

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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.

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I used Cauchy's inequality to get the results.

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I used Titu's Lemma

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

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

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1 and 6 have answers there are infinite solutions

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i will try

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question no. 5 is also very easy.

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

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

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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...

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Thanks a ton!!!!! :)

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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 :)

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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

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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?

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

@Satvik Golechha too, please note this.

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

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please give me the proof of weierstrauss product inequality

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Here is your link :)

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what is the jensens inequality?

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What is the answer to question 5?

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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?

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

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pls explain how to solve the 4th one

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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. )

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How to solve the 2nd question?

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where can I find answer key

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