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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, 2 months ago

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

Harsh Shrivastava - 1 year, 2 months ago

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Done

Naitik Sanghavi - 1 year, 2 months ago

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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

Hrithik Nambiar - 1 year ago

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I got selected for inmo !!!!

Easha Manideep D - 1 year ago

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

Mayank Jha - 1 year ago

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You may post the inmo practice board note.

Only about 1 month left!

Harsh Shrivastava - 1 year ago

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I didn't appear RMO, sorry.

Swapnil Das - 1 year ago

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

Alan Joel - 1 year ago

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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 - 1 year ago

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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.

Anoushka Gupta - 1 year, 1 month 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 year, 1 month ago

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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!

Naitik Sanghavi - 1 year, 1 month ago

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

Easha Manideep D - 1 year, 1 month ago

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Did any one get the 6th one ?

Aryan Saxena - 1 year, 1 month ago

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Is 17 the only possible answer for q3 ?

Aryan Saxena - 1 year, 1 month ago

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Yes.

Harsh Shrivastava - 1 year, 1 month ago

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All the best for your RMO !!!

Alan Joel - 1 year, 1 month ago

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@Alan Joel Thanks!Same to you.

Harsh Shrivastava - 1 year, 1 month ago

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

Easha Manideep D - 1 year, 1 month 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 year, 1 month ago

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Yeah, the answer is zero

Alan Joel - 1 year, 1 month ago

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1st one is done. help for q6.

Mayank Jha - 1 year, 1 month ago

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

Easha Manideep D - 1 year, 1 month ago

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

Brilliant Member - 1 year, 1 month ago

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Did you get any solutions

Easha Manideep D - 1 year, 1 month ago

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

Brilliant Member - 1 year, 1 month ago

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@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 - 1 year, 1 month ago

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@Naitik Sanghavi Yeah I guess

Easha Manideep D - 1 year, 1 month ago

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@Naitik Sanghavi See here

Brilliant Member - 1 year, 1 month ago

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Can someone provide a hint/solution to problem 6?

Brilliant Member - 1 year, 1 month ago

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

Easha Manideep D - 1 year, 1 month ago

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

Easha Manideep D - 1 year, 1 month ago

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

Brilliant Member - 1 year, 1 month ago

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

Easha Manideep D - 1 year, 1 month ago

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Yeah why not.

Easha Manideep D - 1 year, 1 month ago

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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 year, 2 months ago

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Should be greater than 50.

Harsh Shrivastava - 1 year, 2 months ago

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

Arihant Samar - 1 year, 2 months ago

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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

Alan Joel - 1 year, 2 months ago

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

Alan Joel - 1 year, 2 months ago

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Thanx

Kush Singhal - 1 year, 2 months ago

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Q1 seems to be very easy . . . .

Alan Joel - 1 year, 2 months 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 - 1 year, 2 months ago

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Kush singhal E is intersection of AB with circumcircle of ACD

Easha Manideep D - 1 year, 2 months ago

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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?

Kush Singhal - 1 year, 2 months ago

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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 - 1 year, 2 months ago

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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!

Naitik Sanghavi - 1 year, 2 months ago

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Guys today is INMO may the best get selected

Easha Manideep D - 11 months ago

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please post the solution for second one

Sayantan Dhar - 1 year, 1 month ago

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Can someone give solution to q1 and q6

Mayank Jha - 1 year, 1 month ago

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Join I and C in first one

Easha Manideep D - 1 year, 1 month ago

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This year's paper is very easy.

I have my rmo on 16th.

Harsh Shrivastava - 1 year, 2 months ago

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

Brilliant Member - 1 year, 2 months ago

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try question no. 4 of Delhi zone

Sayantan Saha - 1 year, 2 months ago

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@Sayantan Saha I got it

Alan Joel - 1 year, 2 months ago

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

Sayantan Saha - 1 year, 2 months ago

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@Sayantan Saha I got 1

Alan Joel - 1 year, 1 month ago

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@Alan Joel I took a+b+c=1

Sayantan Saha - 1 year, 1 month ago

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When is your rmo?

Harsh Shrivastava - 1 year, 2 months ago

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@Harsh Shrivastava Please post Delhi QP!

Harsh Shrivastava - 1 year, 2 months ago

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@Harsh Shrivastava 16th

Brilliant Member - 1 year, 2 months ago

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Where is it posted?

Harsh Shrivastava - 1 year, 2 months ago

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

Brilliant Member - 1 year, 2 months ago

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@Brilliant Member Thanks, the paper definitily looks tough.

Harsh Shrivastava - 1 year, 2 months ago

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@Harsh Shrivastava On AoPS.

Brilliant Member - 1 year, 2 months ago

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@Brilliant Member Give the link bro!

Harsh Shrivastava - 1 year, 2 months ago

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

Naitik Sanghavi - 1 year, 2 months ago

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11th. Wbu?

Harsh Shrivastava - 1 year, 2 months ago

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

Naitik Sanghavi - 1 year, 2 months ago

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@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.

Brilliant Member - 1 year, 1 month ago

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

Naitik Sanghavi - 1 year, 1 month ago

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@Naitik Sanghavi You got how many correct?

Harsh Shrivastava - 1 year, 1 month ago

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

Naitik Sanghavi - 1 year, 1 month ago

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@Naitik Sanghavi What ????

Easha Manideep D - 1 year, 1 month ago

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@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.

Brilliant Member - 1 year, 2 months ago

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@Naitik Sanghavi I have not given nmtc.

Harsh Shrivastava - 1 year, 2 months ago

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