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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
11 months, 2 weeks ago

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Can you please re-post the question 5, it is not clearly stated. · 11 months, 1 week ago

Done · 11 months, 1 week ago

Guys today is INMO may the best get selected · 8 months, 1 week 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 · 9 months, 2 weeks ago

I got selected for inmo !!!! · 9 months, 1 week ago

I suggest pls ask someone who has prior experience of INMO.He could guide all of us in a better way · 9 months, 2 weeks ago

You may post the inmo practice board note.

Only about 1 month left! · 9 months, 2 weeks ago

I didn't appear RMO, sorry. · 9 months, 2 weeks ago

I qualified RMO from Tamilnadu region !!! Can you guys post an INMO practice board soon ? Only around a month left . · 9 months, 2 weeks 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 ? · 9 months, 2 weeks 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. · 11 months, 1 week 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? · 11 months, 1 week 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! · 11 months, 1 week ago

Yeah that's for sure. But don't worry about that first of all be joyful. · 11 months, 1 week ago

Did any one get the 6th one ? · 11 months, 1 week ago

Is 17 the only possible answer for q3 ? · 11 months, 1 week ago

Yes. · 11 months, 1 week ago

All the best for your RMO !!! · 11 months, 1 week ago

Thanks!Same to you. · 11 months, 1 week ago

How many of you people got fifth one ????? · 11 months, 1 week 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 · 11 months, 1 week ago

Yeah, the answer is zero · 11 months, 1 week ago

1st one is done. help for q6. · 11 months, 1 week ago

Guys did anyone get the 3rd one ????? · 11 months, 1 week ago

Hint: a+b+c=0 if a^3+b^3+c^3-3abc=0 · 11 months, 1 week ago

Did you get any solutions · 11 months, 1 week ago

I got only one solution. i.e. n=17 · 11 months, 1 week 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? · 11 months, 1 week ago

Yeah I guess · 11 months, 1 week ago

See here · 11 months, 1 week ago

Can someone provide a hint/solution to problem 6? · 11 months, 1 week ago

I did it but I have no gaurantee whether I am right or not. : ) · 11 months, 1 week ago

Guys check out the telangana paper same as this one. No change · 11 months, 1 week ago

Are we allowed to use Titu's lemma, Chebyshev's inequality,Holder,etc. to solve inequality problems? · 11 months, 1 week ago

But it can be done through AM GM and cauchy inequality : ) · 11 months, 1 week ago

Yeah why not. · 11 months, 1 week 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 !! · 11 months, 1 week ago

Should be greater than 50. · 11 months, 1 week ago

Can someone post a solution to the inequality question number 4 in the delhi paper? · 11 months, 1 week ago

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

@Kush Singhal Here you go . . . . http://www.artofproblemsolving.com/community/c260h1228775s1rmo2016andinmo2017preparation_group 566th post · 11 months, 1 week ago

Thanx · 11 months, 1 week ago

Q1 seems to be very easy . . . . · 11 months, 1 week ago

In the 5th question, it says that K is the reflection of "E" in BC. But what is "E"? Could it be F? · 11 months, 1 week ago

Kush singhal E is intersection of AB with circumcircle of ACD · 11 months, 1 week 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? · 11 months, 1 week 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? · 11 months, 1 week 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! · 11 months, 1 week ago

please post the solution for second one · 11 months ago

Can someone give solution to q1 and q6 · 11 months, 1 week ago

Join I and C in first one · 11 months, 1 week ago

This year's paper is very easy.

I have my rmo on 16th. · 11 months, 1 week ago

Although I didn't try to solve the Delhi Paper, it seems to be quite hard! · 11 months, 1 week ago

try question no. 4 of Delhi zone · 11 months, 1 week ago

I got it · 11 months, 1 week ago

Then post the solution.The answer should be 27/13. · 11 months, 1 week ago

I got 1 · 11 months, 1 week ago

I took a+b+c=1 · 11 months, 1 week ago

When is your rmo? · 11 months, 1 week ago

Please post Delhi QP! · 11 months, 1 week ago

16th · 11 months, 1 week ago

Where is it posted? · 11 months, 1 week ago

Here The paper is posted somewhere in the 600-750th post so you need to keep scrolling down! · 11 months, 1 week ago

Thanks, the paper definitily looks tough. · 11 months, 1 week ago

On AoPS. · 11 months, 1 week ago

Give the link bro! · 11 months, 1 week ago

Haa! Best of luck for your Rmo! BTW in which standard r u? · 11 months, 1 week ago

11th. Wbu? · 11 months, 1 week ago

Me too in 11th! Do u know that, if we clear nmtc 2nd level can we give inmo? · 11 months, 1 week 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. · 11 months, 1 week ago

Thanks for the info!I'm hoping to clear rmo this time if the cutoff is lesser than 60.... · 11 months, 1 week ago

You got how many correct? · 11 months, 1 week ago

3 are correct but 1 is half right I think so! · 11 months, 1 week ago

What ???? · 11 months, 1 week 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. · 11 months, 1 week ago