# 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.

Note by Naitik Sanghavi
1 year, 11 months ago

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

- 1 year, 11 months ago

Done

- 1 year, 11 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

- 1 year, 9 months ago

I got selected for inmo !!!!

- 1 year, 9 months ago

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

- 1 year, 9 months ago

You may post the inmo practice board note.

- 1 year, 9 months ago

I didn't appear RMO, sorry.

- 1 year, 9 months ago

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

- 1 year, 9 months ago

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 ?

- 1 year, 9 months ago

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.

- 1 year, 11 months ago

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?

- 1 year, 11 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!

- 1 year, 11 months ago

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

- 1 year, 11 months ago

Did any one get the 6th one ?

- 1 year, 11 months ago

Is 17 the only possible answer for q3 ?

- 1 year, 11 months ago

Yes.

- 1 year, 11 months ago

All the best for your RMO !!!

- 1 year, 11 months ago

Thanks!Same to you.

- 1 year, 11 months ago

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

- 1 year, 11 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

- 1 year, 11 months ago

- 1 year, 11 months ago

1st one is done. help for q6.

- 1 year, 11 months ago

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

- 1 year, 11 months ago

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

- 1 year, 11 months ago

Did you get any solutions

- 1 year, 11 months ago

I got only one solution. i.e. n=17

- 1 year, 11 months ago

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?

- 1 year, 11 months ago

Yeah I guess

- 1 year, 11 months ago

See here

- 1 year, 11 months ago

Can someone provide a hint/solution to problem 6?

- 1 year, 11 months ago

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

- 1 year, 11 months ago

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

- 1 year, 11 months ago

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

- 1 year, 11 months ago

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

- 1 year, 11 months ago

Yeah why not.

- 1 year, 11 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 !!

- 1 year, 11 months ago

Should be greater than 50.

- 1 year, 11 months ago

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

- 1 year, 11 months ago

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

- 1 year, 11 months ago

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

- 1 year, 11 months ago

Thanx

- 1 year, 11 months ago

Q1 seems to be very easy . . . .

- 1 year, 11 months ago

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

- 1 year, 11 months ago

Kush singhal E is intersection of AB with circumcircle of ACD

- 1 year, 11 months ago

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?

- 1 year, 11 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?

- 1 year, 11 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!

- 1 year, 11 months ago

Guys today is INMO may the best get selected

- 1 year, 8 months ago

please post the solution for second one

- 1 year, 11 months ago

Can someone give solution to q1 and q6

- 1 year, 11 months ago

Join I and C in first one

- 1 year, 11 months ago

This year's paper is very easy.

I have my rmo on 16th.

- 1 year, 11 months ago

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

- 1 year, 11 months ago

try question no. 4 of Delhi zone

- 1 year, 11 months ago

I got it

- 1 year, 11 months ago

Then post the solution.The answer should be 27/13.

- 1 year, 11 months ago

I got 1

- 1 year, 11 months ago

I took a+b+c=1

- 1 year, 11 months ago

- 1 year, 11 months ago

- 1 year, 11 months ago

16th

- 1 year, 11 months ago

Where is it posted?

- 1 year, 11 months ago

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

- 1 year, 11 months ago

Thanks, the paper definitily looks tough.

- 1 year, 11 months ago

On AoPS.

- 1 year, 11 months ago

- 1 year, 11 months ago

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

- 1 year, 11 months ago

11th. Wbu?

- 1 year, 11 months ago

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

- 1 year, 11 months ago

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.

- 1 year, 11 months ago

Thanks for the info!I'm hoping to clear rmo this time if the cutoff is lesser than 60....

- 1 year, 11 months ago

You got how many correct?

- 1 year, 11 months ago

3 are correct but 1 is half right I think so!

- 1 year, 11 months ago

What ????

- 1 year, 11 months ago

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.

- 1 year, 11 months ago

I have not given nmtc.

- 1 year, 11 months ago