RMO 2016 Gujarat Region

Rmo was conducted on 9th October in Gujarat.

1)Let ABC be a right angled triangle with B=90°.Let I be tye incentre of ABC.Let AI extended intersect BC at F.Draw a line perpendicular to AI at I.Let it intersect AC at E.Prove that IE=IF.

2)Let a,b,c be positive reals such that

(a/1+b)+(b/1+c)+(c/1+a)=1

Prove that abc≤1/8.

3)For any natural number n,expressed in base 10,let S(n) denote the sum of all digits of n.Find all natural numbers n such that

n³=8S(n)³+6nS(n)+1.

4)How many 6 digit natural numbers containing only the digits 1,2,3 are there in which 3 occurs exactly twice and the number Is divisible by 9?

5)Let ABC be a right angled triangle with B=90°.Let AD be the bisector of A with D on BC.Let the circumcircle of triangle ABD intersect AC again in E,and let the circumcircle of triangle ABD intersect AC again in F.Let K be the reflection of E in the line BC.Prove that FK=BC.

6)Show that the infinite arithmetic progression (1,4,7,10......) Has infinitely many 3-term subsequences on harmonic progression such that for any two such triples (a,b,c) and (p,q,r) in harmonic progression ,one has a/p≠b/q.

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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}`	$\boxed{123}$

Comments

Can you please re-post the question 5, it is not clearly stated.

Harsh Shrivastava - 4 years, 8 months ago

Done

naitik sanghavi - 4 years, 8 months ago

The fourth question as it is currently given ^ admits a trivial solution of 0 The condition of exactly two 3's and digits only being 1,2,3 forces the digit sum to be bounded between 10 and 14, and so the number can never be divisible by 9... Is there some typing error?

Kush Singhal - 4 years, 8 months ago

No, it was exactly the same as written on question paper. Everyone asked in exam hall if it had any corrections! I was shocked!!! So I think this year our regions cutoff will go high!

I found the maharashtra and goa region question paper along with a solution pdf here: http://www.rmomah.org/ I found the delhi region question paper here: http://rmonorthzone.com/RMO2016Delhi_zone.pdf I'm still searching for the greater mumbai region's question paper...., if anyone knows where it could be posted could you please tell me?

In the 5th question, it says that K is the reflection of "E" in BC. But what is "E"? Could it be F?

Kush singhal E is intersection of AB with circumcircle of ACD

easha manideep d - 4 years, 8 months ago

Q1 seems to be very easy . . . .

Alan Joel - 4 years, 8 months ago

@Kush Singhal Here you go . . . . http://www.artofproblemsolving.com/community/c260h1228775s1rmo2016andinmo2017preparation_group 566th post

Thanx

RMO 2016 Delhi Region http://rmonorthzone.com/RMO2016Delhi_zone.pdf

RMO 2016 Maharashtra and Goa Region Problems https://doc-0o-ao-docs.googleusercontent.com/docs/securesc/ha0ro937gcuc7l7deffksulhg5h7mbp1/5353lnd22lup3lvu0eqb2f0a7tqb7ucf/1476158400000/01355186314162261890//0BzLwescIQp6cbzQ3UG1vMElSb2M?e=download Solutions https://doc-0c-ao-docs.googleusercontent.com/docs/securesc/ha0ro937gcuc7l7deffksulhg5h7mbp1/j1l5kkjqoutlkrq6f1cs5r899sfao56f/1476158400000/01355186314162261890//0BzLwescIQp6cOVBpRXFtaVpOMDA?e=download

RMO 2016 Mumbai Region http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMy85L2Q1NGE2ZjA4ZTY4YWZkZmE5ZDA0OTFlOTBkOGQyNmM1NGMxYTlmLmpwZw==&rn=U2Nhbl8yMDE2MTAwOS1taW4uanBn

Can someone post a solution to the inequality question number 4 in the delhi paper?

Arihant Samar - 4 years, 8 months ago

What do you all think will bebl the cutoff for this region?Last years cutoff was 31 and approx 30-40 in past years !!

Should be greater than 50.

Are we allowed to use Titu's lemma, Chebyshev's inequality,Holder,etc. to solve inequality problems?

A Former Brilliant Member - 4 years, 8 months ago

Yeah why not.

But it can be done through AM GM and cauchy inequality : )

Guys check out the telangana paper same as this one. No change

Can someone provide a hint/solution to problem 6?

I did it but I have no gaurantee whether I am right or not. : )

Guys did anyone get the 3rd one ?????

Hint: a+b+c=0 if a^3+b^3+c^3-3abc=0

Did you get any solutions

@Easha Manideep D – I got only one solution. i.e. n=17

@A Former Brilliant Member – I also got only 17,But I didnt observe a³+b³+c³-3abc=0,I factorized it. I think I will lose marks for that!! anyways, did u get 6th?

@Naitik Sanghavi – See here

@Naitik Sanghavi – Yeah I guess

1st one is done. help for q6.

Mayank Jha - 4 years, 8 months ago

Is q4 answer 0? As the number is divisible by 9 sum of remaining 4 digits +6 must me divisible by 9 but no case is satisfied

manas verma - 4 years, 8 months ago

Yeah, the answer is zero

How many of you people got fifth one ?????

Is 17 the only possible answer for q3 ?

aryan saxena - 4 years, 8 months ago

Yes.

All the best for your RMO !!!

@Alan Joel – Thanks!Same to you.

Did any one get the 6th one ?

Do the RMO answer sheet checkers cut marks for long methods, provided they have a very short method in their answer key? Any idea how many will be selected for INMO this year from Gujarat?

anoushka gupta - 4 years, 8 months ago

Dont know about checking but I think they would cut some marks for that!30-35 will be selected and top 5 girl students extra!

Yeah that's for sure. But don't worry about that first of all be joyful.

Thanks! In the 4th q, I ended up writing "NO no. will be possible.." instead of "0". Idiotic, I know. Let's wait for results now.

Guys I scored 41/102 .My questions were the same as mentioned above even though I belong to a different region. Do I stand a fair chance of qualification ?

Hrithik Nambiar - 4 years, 6 months ago

Hey guys! The RMO 2016 results are out, who all made it to INMO. Pls do create notes and discussion for the upcoming INMO. Thank you. @Harsh Shrivastava @naitik sanghavi @Mayank Jha @Rajdeep Dhingra @Anik Mandal @Swapnil Das

I got selected for inmo !!!!

easha manideep d - 4 years, 6 months ago

I didn't appear RMO, sorry.

Swapnil Das - 4 years, 6 months ago

I qualified RMO from Tamilnadu region !!! Can you guys post an INMO practice board soon ? Only around a month left .

Alan Joel - 4 years, 6 months ago

You may post the inmo practice board note.

Only about 1 month left!

Harsh Shrivastava - 4 years, 6 months ago

I suggest pls ask someone who has prior experience of INMO.He could guide all of us in a better way

Mayank Jha - 4 years, 6 months ago

Can someone give solution to q1 and q6

Join I and C in first one

please post the solution for second one

Sayantan Dhar - 4 years, 8 months ago

Guys today is INMO may the best get selected

easha manideep d - 4 years, 5 months ago

This year's paper is very easy.

I have my rmo on 16th.

Haa! Best of luck for your Rmo! BTW in which standard r u?

11th. Wbu?

@Harsh Shrivastava – Me too in 11th! Do u know that, if we clear nmtc 2nd level can we give inmo?

@Naitik Sanghavi – I have not given nmtc.

@Naitik Sanghavi – I have the same doubt. Last year I think that top 30 or so would appear for INMO but nothing about it has been mentioned this year so I guess that the policy has been stopped. I will call them and ask soon.

@Naitik Sanghavi – Just called them up. They said that you can qualify for INMO only through RMO and you get only cash awards and scholarship if you clear 2nd level.

@A Former Brilliant Member – Thanks for the info!I'm hoping to clear rmo this time if the cutoff is lesser than 60....

@Naitik Sanghavi – You got how many correct?

@Harsh Shrivastava – 3 are correct but 1 is half right I think so!

@Naitik Sanghavi – What ????

Although I didn't try to solve the Delhi Paper, it seems to be quite hard!

Where is it posted?

@Harsh Shrivastava – On AoPS.

@A Former Brilliant Member – Give the link bro!

@Harsh Shrivastava – Here The paper is posted somewhere in the 600-750th post so you need to keep scrolling down!

@A Former Brilliant Member – Thanks, the paper definitily looks tough.

When is your rmo?

@Harsh Shrivastava – 16th

@Harsh Shrivastava – Please post Delhi QP!

try question no. 4 of Delhi zone

Sayantan Saha - 4 years, 8 months ago

@Sayantan Saha – I got it

@Alan Joel – Then post the solution.The answer should be 27/13.

@Sayantan Saha – I got 1

@Alan Joel – I took a+b+c=1

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}`	$\boxed{123}$