RMO Paper 2013 of Maharashtra and Goa region

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

(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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`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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

arun patil - 7 years, 6 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 1

Aditya Raut - 7 years, 6 months ago

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

Anuj Shikarkhane - 6 years, 5 months ago

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

Aditya Raut - 6 years, 5 months ago

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

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 - 7 years, 6 months ago

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

which books did you use for RMO? @Aditya Raut

Dev Sharma - 5 years, 9 months ago

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

AVNISH GARG - 3 years, 8 months ago

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

Ram Keswani - 1 year, 10 months ago

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

Hrushikesh bhope - 7 years, 6 months ago

Answer to 1st question:-

$m \in \mathbb{N}-$ {4,p}

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

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

how did we solve question number 4

Rahul Rawat - 5 years, 5 months ago

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.........

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.

I used Cauchy's inequality to get the results.

I used Titu's Lemma

Ranjana Kasangeri - 7 years, 6 months ago

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

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

1 and 6 have answers there are infinite solutions

Anurag Ramachandran - 3 years, 9 months ago

i will try

Sanskruti Lohia - 7 years, 6 months ago

question no. 5 is also very easy.

akhilesh agrawal - 7 years, 6 months ago

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

Hrishikesh Dani - 7 years, 5 months ago

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

Krishna Ar - 6 years, 10 months ago

DONE .... $\color{#D61F06}{T} \color{#EC7300}{H} \color{limegreen}{A} \color{#20A900}{N} \color{#3D99F6}{K} \color{#69047E}{S} \color{#E81990}{S} \color{darkred}{S}$ .... 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 - 6 years, 10 months ago

Thanks a ton!!!!! :)

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

Inequalities

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

App. Comb

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

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

PSS PSS

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

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

$\color{#D61F06}{\textbf{IMO problems}}$ (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 $\color{#3D99F6}{\textbf{TRICK}}$ doesn't $\color{#3D99F6}{\textbf{CLICK}}$ 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 ₹ $3379/-$ , 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 – bro thanks for giving such a useful website

akash deep - 6 years, 10 months ago

@Akash Deep – Anytime ! Wanna be friends ? Gmail- $\color{#3D99F6}{\text{[email protected] }}$

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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 – Anytime, I am there for anyone who wants to be friends! By the way, please send me email if you want, at $\color{#3D99F6}{\textbf{[email protected]}}$ instead of '[email protected]', 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 – Sure..thanks for notifying this too :P :D

please give me the proof of weierstrauss product inequality

dipak bhole - 6 years, 6 months ago

Here is your link :)

Aditya Raut - 6 years, 6 months ago

what is the jensens inequality?

dipak bhole - 6 years, 1 month ago

What is the answer to question 5?

Shashank Rammoorthy - 5 years, 6 months ago

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 - 4 years, 9 months ago

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

A Former Brilliant Member - 4 years, 3 months ago

pls explain how to solve the 4th one

neelmadhav sahu - 3 years, 8 months ago

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

Kinshuk Bansal - 1 year, 7 months ago

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. )

How to solve the 2nd question?

Aditya Patil - 7 years, 5 months ago

where can I find answer key

Gopalkrishna Nayak Pangal - 6 years, 6 months ago

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}`	$\boxed{123}$