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.

## Comments

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TopNewestThanks! 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. – Anoushka Gupta · 1 week, 1 day ago

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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 · 1 week, 2 days ago

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– Naitik Sanghavi · 1 week, 2 days 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!Log in to reply

– Easha Manideep D · 1 week, 2 days ago

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

Did any one get the 6th one ? – Aryan Saxena · 1 week, 3 days ago

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Is 17 the only possible answer for q3 ? – Aryan Saxena · 1 week, 3 days ago

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– Harsh Shrivastava · 1 week, 3 days ago

Yes.Log in to reply

– Alan Joel · 1 week, 3 days ago

All the best for your RMO !!!Log in to reply

– Harsh Shrivastava · 1 week, 3 days ago

Thanks!Same to you.Log in to reply

How many of you people got fifth one ????? – Easha Manideep D · 1 week, 3 days ago

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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 · 1 week, 3 days ago

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– Alan Joel · 1 week, 3 days ago

Yeah, the answer is zeroLog in to reply

1st one is done. help for q6. – Mayank Jha · 1 week, 4 days ago

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Guys did anyone get the 3rd one ????? – Easha Manideep D · 1 week, 4 days ago

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– Svatejas Shivakumar · 1 week, 4 days ago

Hint: a+b+c=0 if a^3+b^3+c^3-3abc=0Log in to reply

– Easha Manideep D · 1 week, 4 days ago

Did you get any solutionsLog in to reply

– Svatejas Shivakumar · 1 week, 4 days ago

I got only one solution. i.e. n=17Log in to reply

– Naitik Sanghavi · 1 week, 4 days 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?Log in to reply

– Easha Manideep D · 1 week, 3 days ago

Yeah I guessLog in to reply

here – Svatejas Shivakumar · 1 week, 4 days ago

SeeLog in to reply

Can someone provide a hint/solution to problem 6? – Svatejas Shivakumar · 1 week, 5 days ago

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– Easha Manideep D · 1 week, 5 days ago

I did it but I have no gaurantee whether I am right or not. : )Log in to reply

Guys check out the telangana paper same as this one. No change – Easha Manideep D · 1 week, 5 days ago

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Are we allowed to use Titu's lemma, Chebyshev's inequality,Holder,etc. to solve inequality problems? – Svatejas Shivakumar · 1 week, 5 days ago

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– Easha Manideep D · 1 week, 5 days ago

But it can be done through AM GM and cauchy inequality : )Log in to reply

– Easha Manideep D · 1 week, 5 days ago

Yeah why not.Log in to reply

What do you all think will bebl the cutoff for this region?Last years cutoff was 31 and approx 30-40 in past years !! – Naitik Sanghavi · 1 week, 5 days ago

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– Harsh Shrivastava · 1 week, 5 days ago

Should be greater than 50.Log in to reply

Can someone post a solution to the inequality question number 4 in the delhi paper? – Arihant Samar · 1 week, 6 days ago

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

2016Delhi_zone.pdfRMO 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=downloadRMO 2016 Mumbai Region http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMy85L2Q1NGE2ZjA4ZTY4YWZkZmE5ZDA0OTFlOTBkOGQyNmM1NGMxYTlmLmpwZw==&rn=U2Nhbl8yMDE2MTAwOS1taW4uanBn – Alan Joel · 2 weeks ago

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@Kush Singhal Here you go . . . . http://www.artofproblemsolving.com/community/c260h1228775s1

rmo2016andinmo2017preparation_group 566th post – Alan Joel · 2 weeks agoLog in to reply

– Kush Singhal · 2 weeks ago

ThanxLog in to reply

Q1 seems to be very easy . . . . – Alan Joel · 2 weeks ago

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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 · 2 weeks ago

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– Easha Manideep D · 1 week, 6 days ago

Kush singhal E is intersection of AB with circumcircle of ACDLog in to reply

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/RMO

2016Delhi_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? – Kush Singhal · 2 weeks agoLog in to reply

Can you please re-post the question 5, it is not clearly stated. – Harsh Shrivastava · 2 weeks ago

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– Naitik Sanghavi · 2 weeks ago

DoneLog in to reply

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 · 2 weeks ago

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– Naitik Sanghavi · 2 weeks 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!Log in to reply

please post the solution for second one – Sayantan Dhar · 6 days, 18 hours ago

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Can someone give solution to q1 and q6 – Mayank Jha · 1 week, 4 days ago

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– Easha Manideep D · 1 week, 4 days ago

Join I and C in first oneLog in to reply

This year's paper is very easy.

I have my rmo on 16th. – Harsh Shrivastava · 2 weeks ago

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– Svatejas Shivakumar · 2 weeks ago

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

– Sayantan Saha · 2 weeks ago

try question no. 4 of Delhi zoneLog in to reply

– Alan Joel · 2 weeks ago

I got itLog in to reply

– Sayantan Saha · 1 week, 6 days ago

Then post the solution.The answer should be 27/13.Log in to reply

– Alan Joel · 1 week, 3 days ago

I got 1Log in to reply

– Sayantan Saha · 1 week, 3 days ago

I took a+b+c=1Log in to reply

– Harsh Shrivastava · 2 weeks ago

When is your rmo?Log in to reply

– Harsh Shrivastava · 2 weeks ago

Please post Delhi QP!Log in to reply

– Svatejas Shivakumar · 2 weeks ago

16thLog in to reply

– Harsh Shrivastava · 2 weeks ago

Where is it posted?Log in to reply

Here The paper is posted somewhere in the 600-750th post so you need to keep scrolling down! – Svatejas Shivakumar · 2 weeks ago

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– Harsh Shrivastava · 2 weeks ago

Thanks, the paper definitily looks tough.Log in to reply

– Svatejas Shivakumar · 2 weeks ago

On AoPS.Log in to reply

– Harsh Shrivastava · 2 weeks ago

Give the link bro!Log in to reply

– Naitik Sanghavi · 2 weeks ago

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

– Harsh Shrivastava · 2 weeks ago

11th. Wbu?Log in to reply

– Naitik Sanghavi · 2 weeks ago

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

– Svatejas Shivakumar · 1 week, 5 days 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.Log in to reply

– Naitik Sanghavi · 1 week, 5 days ago

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

– Harsh Shrivastava · 1 week, 4 days ago

You got how many correct?Log in to reply

– Naitik Sanghavi · 1 week, 4 days ago

3 are correct but 1 is half right I think so!Log in to reply

– Easha Manideep D · 1 week, 4 days ago

What ????Log in to reply

– Svatejas Shivakumar · 2 weeks 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.Log in to reply

– Harsh Shrivastava · 2 weeks ago

I have not given nmtc.Log in to reply