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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
3 months, 1 week ago

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Can you please re-post the question 5, it is not clearly stated. Harsh Shrivastava · 3 months, 1 week ago

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@Harsh Shrivastava Done Naitik Sanghavi · 3 months, 1 week ago

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Guys today is INMO may the best get selected Easha Manideep D · 5 days, 23 hours 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 month, 2 weeks ago

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@Hrithik Nambiar I got selected for inmo !!!! Easha Manideep D · 1 month, 1 week ago

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@Hrithik Nambiar I suggest pls ask someone who has prior experience of INMO.He could guide all of us in a better way Mayank Jha · 1 month, 1 week ago

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@Hrithik Nambiar You may post the inmo practice board note.

Only about 1 month left! Harsh Shrivastava · 1 month, 2 weeks ago

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@Hrithik Nambiar I didn't appear RMO, sorry. Swapnil Das · 1 month, 2 weeks ago

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@Swapnil Das I qualified RMO from Tamilnadu region !!! Can you guys post an INMO practice board soon ? Only around a month left . Alan Joel · 1 month, 1 week 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 month, 2 weeks 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 · 3 months 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 · 3 months ago

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@Anoushka Gupta 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 · 3 months ago

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@Naitik Sanghavi Yeah that's for sure. But don't worry about that first of all be joyful. Easha Manideep D · 3 months ago

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

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

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@Aryan Saxena Yes. Harsh Shrivastava · 3 months, 1 week ago

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@Harsh Shrivastava All the best for your RMO !!! Alan Joel · 3 months, 1 week ago

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@Alan Joel Thanks!Same to you. Harsh Shrivastava · 3 months, 1 week ago

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

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@Manas Verma Yeah, the answer is zero Alan Joel · 3 months, 1 week ago

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1st one is done. help for q6. Mayank Jha · 3 months, 1 week ago

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

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@Easha Manideep D Hint: a+b+c=0 if a^3+b^3+c^3-3abc=0 Svatejas Shivakumar · 3 months, 1 week ago

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@Svatejas Shivakumar Did you get any solutions Easha Manideep D · 3 months, 1 week ago

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@Easha Manideep D I got only one solution. i.e. n=17 Svatejas Shivakumar · 3 months, 1 week ago

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@Svatejas Shivakumar 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 · 3 months, 1 week ago

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@Naitik Sanghavi Yeah I guess Easha Manideep D · 3 months, 1 week ago

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@Naitik Sanghavi See here Svatejas Shivakumar · 3 months, 1 week ago

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Can someone provide a hint/solution to problem 6? Svatejas Shivakumar · 3 months, 1 week ago

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@Svatejas Shivakumar I did it but I have no gaurantee whether I am right or not. : ) Easha Manideep D · 3 months, 1 week ago

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Guys check out the telangana paper same as this one. No change Easha Manideep D · 3 months, 1 week ago

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Are we allowed to use Titu's lemma, Chebyshev's inequality,Holder,etc. to solve inequality problems? Svatejas Shivakumar · 3 months, 1 week ago

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@Svatejas Shivakumar But it can be done through AM GM and cauchy inequality : ) Easha Manideep D · 3 months, 1 week ago

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@Svatejas Shivakumar Yeah why not. Easha Manideep D · 3 months, 1 week 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 · 3 months, 1 week ago

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@Naitik Sanghavi Should be greater than 50. Harsh Shrivastava · 3 months, 1 week ago

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Can someone post a solution to the inequality question number 4 in the delhi paper? Arihant Samar · 3 months, 1 week 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 · 3 months, 1 week ago

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@Kush Singhal Here you go . . . . http://www.artofproblemsolving.com/community/c260h1228775s1rmo2016andinmo2017preparation_group 566th post Alan Joel · 3 months, 1 week ago

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@Alan Joel Thanx Kush Singhal · 3 months, 1 week ago

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Q1 seems to be very easy . . . . Alan Joel · 3 months, 1 week 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 · 3 months, 1 week ago

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@Kush Singhal Kush singhal E is intersection of AB with circumcircle of ACD Easha Manideep D · 3 months, 1 week 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 · 3 months, 1 week 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 · 3 months, 1 week ago

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@Kush Singhal 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 · 3 months, 1 week ago

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please post the solution for second one Sayantan Dhar · 3 months ago

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Can someone give solution to q1 and q6 Mayank Jha · 3 months, 1 week ago

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@Mayank Jha Join I and C in first one Easha Manideep D · 3 months, 1 week ago

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This year's paper is very easy.

I have my rmo on 16th. Harsh Shrivastava · 3 months, 1 week ago

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@Harsh Shrivastava Although I didn't try to solve the Delhi Paper, it seems to be quite hard! Svatejas Shivakumar · 3 months, 1 week ago

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@Svatejas Shivakumar try question no. 4 of Delhi zone Sayantan Saha · 3 months, 1 week ago

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@Sayantan Saha I got it Alan Joel · 3 months, 1 week ago

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@Alan Joel Then post the solution.The answer should be 27/13. Sayantan Saha · 3 months, 1 week ago

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@Sayantan Saha I got 1 Alan Joel · 3 months, 1 week ago

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@Alan Joel I took a+b+c=1 Sayantan Saha · 3 months, 1 week ago

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@Svatejas Shivakumar When is your rmo? Harsh Shrivastava · 3 months, 1 week ago

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@Harsh Shrivastava Please post Delhi QP! Harsh Shrivastava · 3 months, 1 week ago

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@Harsh Shrivastava 16th Svatejas Shivakumar · 3 months, 1 week ago

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@Svatejas Shivakumar Where is it posted? Harsh Shrivastava · 3 months, 1 week 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! Svatejas Shivakumar · 3 months, 1 week ago

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@Svatejas Shivakumar Thanks, the paper definitily looks tough. Harsh Shrivastava · 3 months, 1 week ago

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@Harsh Shrivastava On AoPS. Svatejas Shivakumar · 3 months, 1 week ago

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@Svatejas Shivakumar Give the link bro! Harsh Shrivastava · 3 months, 1 week ago

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@Harsh Shrivastava Haa! Best of luck for your Rmo! BTW in which standard r u? Naitik Sanghavi · 3 months, 1 week ago

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@Naitik Sanghavi 11th. Wbu? Harsh Shrivastava · 3 months, 1 week 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 · 3 months, 1 week 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. Svatejas Shivakumar · 3 months, 1 week ago

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@Svatejas Shivakumar Thanks for the info!I'm hoping to clear rmo this time if the cutoff is lesser than 60.... Naitik Sanghavi · 3 months, 1 week ago

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@Naitik Sanghavi You got how many correct? Harsh Shrivastava · 3 months, 1 week ago

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@Harsh Shrivastava 3 are correct but 1 is half right I think so! Naitik Sanghavi · 3 months, 1 week ago

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@Naitik Sanghavi What ???? Easha Manideep D · 3 months, 1 week 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. Svatejas Shivakumar · 3 months, 1 week ago

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@Naitik Sanghavi I have not given nmtc. Harsh Shrivastava · 3 months, 1 week ago

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