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RMO 2016 Karnataka Region(16th Oct)

  1. Let \(ABC\) be a triangle and \(D\) be the mid-point of \(BC\). Suppose the angle bisector of \(\angle ADC\) is tangent to the circumcircle of triangle \(ABD\) at \(D\). Prove that \(\angle A=90^{\circ}\).

  2. Let \(a,b,c\) be three distinct positive real numbers such that \(abc=1\). Prove that \(\dfrac{a^3}{(a-b)(a-c)}+\dfrac{b^3}{(b-c)(b-a)}+\dfrac{c^3}{(c-a)(c-b)} > 3\).

  3. Let \(a,b,c,d,e,d,e,f\) be positive integers such that \(\dfrac a b < \dfrac c d < \dfrac e f\). Suppose \(af-be=-1\). Show that \(d \geq b+f\).

  4. There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integer such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in at least one common language, and no language is common to all \(100\) countries, what is the minimum possible value of \(n\)?

  5. Let \(ABC\) be a right-angled triangle with \(\angle B=90^{\circ}\). Let \(I\) be the incentre if \(ABC\). Extend \(AI\) and \(CI\); let them intersect \(BC\) in \(D\) and \(AB\) in \(E\) respectively. Draw a line perpendicular to \(AI\) at \(I\) to meet \(AC\) in \(J\), draw a line perpendicular to \(CI\) at \(I\) to meet \(AC\) at \(K\). Suppose \(DJ=EK\). Prove that \(BA=BC\).

6.(a). Given any natural number \(N\), prove that there exists a strictly increasing sequence of \(N\) positive integers in harmonic progression.

(b). Prove that there cannot exist a strictly increasing infinite sequence of positive integers which is in harmonic progression.

Note by Svatejas Shivakumar
5 months, 1 week ago

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solutions are now uploaded on resonance. Svatejas Shivakumar · 5 months, 1 week ago

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@Svatejas Shivakumar Have you come to know your result? Sayantan Saha · 4 months, 3 weeks ago

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@Svatejas Shivakumar What is your rmo score? Sayantan Saha · 4 months, 3 weeks ago

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@Sayantan Saha I registered through FIITJEE so I will receive my marks when I have my next class(tomorrow). Svatejas Shivakumar · 4 months, 3 weeks ago

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@Sayantan Saha What is yours? Mine is very bad. Kushagra Sahni · 4 months, 3 weeks ago

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@Kushagra Sahni Whats ur score pLS tell! Kaustubh Miglani · 4 months, 3 weeks ago

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@Kaustubh Miglani It's below 10. Almost everyone from Fiitjee south Delhi including sankalp got below 10. What's your score? Kushagra Sahni · 4 months, 3 weeks ago

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@Kushagra Sahni Was Delhi RMO paper different from that of Chhattisgarh Region?? Was it more difficult?? Saurabh Mallik · 4 months, 3 weeks ago

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@Saurabh Mallik It was of INMO level this time. Almost everyone was shocked this time. Kushagra Sahni · 4 months, 3 weeks ago

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@Kushagra Sahni Same here as well. Saurabh Mallik · 4 months, 3 weeks ago

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@Saurabh Mallik It was different plus exam pressure also makes a huge difference. Harsh Shrivastava · 4 months, 3 weeks ago

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@Harsh Shrivastava Yes and if the paper is like that we get more scared and I realised I could have got 5-6 marks more had I been not depressed after looking at the paper. I lost all my confidence after looking at the paper. Kushagra Sahni · 4 months, 3 weeks ago

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@Kushagra Sahni How many questions did you solve Harsh?? Saurabh Mallik · 4 months, 3 weeks ago

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@Kaustubh Miglani Our(West Bengal's) result has not come out. Sayantan Saha · 4 months, 3 weeks ago

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@Kushagra Sahni I am from WB region. Result of our region has not come out. Sayantan Saha · 4 months, 3 weeks ago

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@Svatejas Shivakumar It was just 5 minutes after the exam was over LOL Vicky Vignesh · 5 months, 1 week ago

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I Did the first one now in 10 min Meera Somani · 3 weeks, 3 days ago

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@Meera Somani It's easy.Which class for you? And did you write RMO? Anik Mandal · 3 weeks, 3 days ago

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@Rohit Camfar I am studying for boards now,So probably later.. Anik Mandal · 3 weeks, 3 days ago

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Solution 1.link text Rohit Camfar · 3 months, 1 week ago

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How many of you all are selected. Mayank Jha · 3 months, 3 weeks ago

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@Mayank Jha I am selected from Chattisgarh region. Harsh Shrivastava · 3 months, 3 weeks ago

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How much scores pals. Mayank Jha · 4 months, 1 week ago

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@Mayank Jha 51 Wasif Jawad Hussain · 4 months, 1 week ago

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@Wasif Jawad Hussain Yours ??? Wasif Jawad Hussain · 4 months, 1 week ago

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@Wasif Jawad Hussain Wasif how much you scored in Inmo? Mayank Jha · 1 month, 1 week ago

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@Wasif Jawad Hussain Wasif what is the highest score in jharkhand Mayank Jha · 3 months, 3 weeks ago

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@Wasif Jawad Hussain Yours? Wasif Jawad Hussain · 4 months, 1 week ago

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@Wasif Jawad Hussain How much is the highest score in your city? Mayank Jha · 4 months, 1 week ago

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@Mayank Jha What is the highest score you have heard till now Mayank Jha · 4 months, 1 week ago

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@Mayank Jha Some other scores are 48,32... Wasif Jawad Hussain · 4 months, 1 week ago

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@Wasif Jawad Hussain What is your kvpy expected score Mayank Jha · 4 months, 1 week ago

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@Mayank Jha 50-55 as per old answer key Wasif Jawad Hussain · 4 months, 1 week ago

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@Wasif Jawad Hussain 46.Can you tell how much chances I have to qualify Mayank Jha · 4 months, 1 week ago

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@Mayank Jha By now i have heard 51(mine),34,41,...and waiting for others . do you know anyone with 60+ score ? Wasif Jawad Hussain · 4 months, 1 week ago

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@Wasif Jawad Hussain You have not heard above 60 score till now Mayank Jha · 4 months, 1 week ago

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@Wasif Jawad Hussain What are the scores from jamshedpur ? Wasif Jawad Hussain · 4 months, 1 week ago

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@Wasif Jawad Hussain Sorry I have no idea till now.Can you give me your mail address Mayank Jha · 4 months, 1 week ago

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@Mayank Jha What are the scores from jamshedpur ????? Wasif Jawad Hussain · 4 months ago

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@Wasif Jawad Hussain When will the result of Rmo be declared? Mayank Jha · 4 months ago

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@Wasif Jawad Hussain Some scores are 15,25 many not received any emails. Mayank Jha · 4 months ago

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@Wasif Jawad Hussain Sorry wasif no information till now.my friends do not check their mail. Mayank Jha · 4 months ago

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@Svatejas Shivakumar @Sharky Kesa Can either of you help me verify the phrasing of Q4? Is it "exactly one common language" or "at least one common language"?

It makes a huge difference in appoaching the problem. Calvin Lin Staff · 5 months, 1 week ago

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@Calvin Lin It is atleast one common language. Anik Mandal · 5 months, 1 week ago

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@Anik Mandal Thanks. I've edited the question accordingly. Calvin Lin Staff · 5 months ago

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Anyone with 6 th done Aakash Khandelwal · 5 months, 1 week ago

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@Aakash Khandelwal Both of \(6\)'s subdivions are available in Brilliant.

Grasp here and here Vicky Vignesh · 5 months, 1 week ago

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20 is the answer for the 4th one Mayank Jha · 5 months, 1 week ago

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Is the following solution correct for P2?

WLOG \(a>b>c\) Multiplying both sides by \((a-b)(b-c)(c-a)\), it suffices to prove \(\sum a^3(b-c) \ge 3(a-b)(b-c)(c-a)\).

Expanding both sides and simplifying , this reduces to prove \(\sum (a^3b+3a^2b) \ge \sum( ab^3+3ab^2)\). Using \(abc=1\) this simplifies to show \[\sum \frac{a^2+3a}{c} \ge \sum \frac{a^2+3a}{b}\]

But this is just rearrangement on \[a^3+3a>b^3+3b>c^3+3c\] and \[1/a < 1/b < 1/c\] AceKnight Trogh · 5 months, 1 week ago

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what is the answer of the 4 th one Abhishek Alva · 5 months, 1 week ago

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The same question paper was asked in jharkand . Being in eleventh is solving three correctly , enough to qualify ? Wasif Jawad Hussain · 5 months, 1 week ago

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Has anyone perfectly solved Q.4?? Saurabh Mallik · 5 months, 1 week ago

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@Sharky Kesa The Farey sequence relates to question 3. Calvin Lin Staff · 5 months, 1 week ago

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@Calvin Lin Yeah, sorry, looked at the wrong question. Question 4 looks like Pigeonhole Principle. Sharky Kesa · 5 months, 1 week ago

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For q2 since a,b,c are distinct WLOG a>b>c.Take a=c+y,b=c+x,and replace a,b with the above values in lhs and on solving you get finally lhs as 3c+x+y which is nothing but a+b+c,now apply am-gm,to prove it. Mayank Jha · 5 months, 1 week ago

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Terms are [a^3/{3(a-b)(a-c)}]+a-b/(3^{5/2})+a-c/(3^{5/2})>=a/3 Manas Verma · 5 months, 1 week ago

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@Manas Verma Better use the formatting guide to write the expression.

\(\frac{a^{3}}{3(a-b)(a-c)}+\frac{a-b}{3^{5/2}}+\frac{a-c}{3^{5/2}} \geq \frac{a}{3}\)

That looks better and much understandable too! Saurabh Mallik · 5 months, 1 week ago

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@Manas Verma *a-b is also in bracket.sane with a-c Manas Verma · 5 months, 1 week ago

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Terms can be negative but they are eventually cancelled so we can apply am-gm Manas Verma · 5 months, 1 week ago

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@Svatejas Shivakumar Could you post the solution to problem 6? Milind Prabhu · 5 months, 1 week ago

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I am from jharkhand.I have solved q1,2,5,6.solution for q4 please Mayank Jha · 5 months, 1 week ago

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@Mayank Jha You are in which class and did you got dps ranchi as exam centre ? Wasif Jawad Hussain · 5 months, 1 week ago

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@Wasif Jawad Hussain I am from Jamshedpur,in class 11 Mayank Jha · 5 months, 1 week ago

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Same paper was for chhattisgarh also! Manas Verma · 5 months, 1 week ago

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Apply am -gm for these a^3/3(a-b)(a-c)+(a-b)/3^(5/2)+(a-c)/3^(5/2) Which is>=a/3 Now write same expressions for b and c and add all three We get in rhs- a+b+c/3>=1(am-gm) Hence proved :) Manas Verma · 5 months, 1 week ago

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@Manas Verma You cannot apply AM GM Its not given that a-b,b-c,c-a are +ve Kaustubh Miglani · 5 months, 1 week ago

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@Naitik Sanghavi Yeah you are right Kaustubh Miglani · 5 months, 1 week ago

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@Kaustubh Miglani But still denominator of second term is -ve

BTW,Rank In Technothlon? Kaustubh Miglani · 5 months, 1 week ago

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@Manas Verma can u put brackets please Abhishek Alva · 5 months, 1 week ago

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you can solve the second one by using am gm inequality Abhishek Alva · 5 months, 1 week ago

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@Abhishek Alva Can you post the solution? Also when does equality occur? Svatejas Shivakumar · 5 months, 1 week ago

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@Svatejas Shivakumar Which questions did u do? Kaustubh Miglani · 5 months, 1 week ago

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@Kaustubh Miglani Attempted all. Got 1,5,6 completely. Don't know the answer of 4. Did some progress in 1 and 2. Svatejas Shivakumar · 5 months, 1 week ago

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@Svatejas Shivakumar I got 3 questions fully correct and 1 question 25% corrrect Kaustubh Miglani · 5 months, 1 week ago

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@Svatejas Shivakumar Dont u think q-1 is halwa? NCERT Question!!! Kaustubh Miglani · 5 months, 1 week ago

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@Kaustubh Miglani Really!! Though it was very easy but don't think it could be an NCERT question. Saurabh Mallik · 5 months, 1 week ago

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@Kaustubh Miglani Yes even 5 was very easy. Svatejas Shivakumar · 5 months, 1 week ago

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@Svatejas Shivakumar Can you please post the complete solution of Q.5?? Saurabh Mallik · 5 months, 1 week ago

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@Kaustubh Miglani Yes very easy everyone was able to solve it. Did you do Q.3? Kushagra Sahni · 5 months, 1 week ago

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@Kushagra Sahni Yeah did it Though I spent 55 mins on this one Kaustubh Miglani · 5 months, 1 week ago

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@Kaustubh Miglani Ok it took me also around 30-40 mins. How was RMO? Kushagra Sahni · 5 months, 1 week ago

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@Kushagra Sahni I gave both RMO and GMO

Both went good Kaustubh Miglani · 5 months, 1 week ago

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@Svatejas Shivakumar It is better to evaluate it to be a+b+c Kaustubh Miglani · 5 months, 1 week ago

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@Svatejas Shivakumar It is better if u factorise given expression It is a+b+c Cdsc Dsc · 5 months, 1 week ago

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All these problems except 3rd one is posted in this brilliant with solutions. Please refer the medium rated problems. Vicky Vignesh · 5 months, 1 week ago

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