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 \(\angle A=90^{\circ}\).

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

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 \(d \geq b+f\).

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 \(n\)?

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.

## Comments

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

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– Sayantan Saha · 4 months, 3 weeks ago

Have you come to know your result?Log in to reply

– Sayantan Saha · 4 months, 3 weeks ago

What is your rmo score?Log in to reply

– Svatejas Shivakumar · 4 months, 3 weeks ago

I registered through FIITJEE so I will receive my marks when I have my next class(tomorrow).Log in to reply

– Kushagra Sahni · 4 months, 3 weeks ago

What is yours? Mine is very bad.Log in to reply

– Kaustubh Miglani · 4 months, 3 weeks ago

Whats ur score pLS tell!Log in to reply

– Kushagra Sahni · 4 months, 3 weeks ago

It's below 10. Almost everyone from Fiitjee south Delhi including sankalp got below 10. What's your score?Log in to reply

– Saurabh Mallik · 4 months, 3 weeks ago

Was Delhi RMO paper different from that of Chhattisgarh Region?? Was it more difficult??Log in to reply

– Kushagra Sahni · 4 months, 3 weeks ago

It was of INMO level this time. Almost everyone was shocked this time.Log in to reply

– Saurabh Mallik · 4 months, 3 weeks ago

Same here as well.Log in to reply

– Harsh Shrivastava · 4 months, 3 weeks ago

It was different plus exam pressure also makes a huge difference.Log in to reply

– Kushagra Sahni · 4 months, 3 weeks ago

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.Log in to reply

– Saurabh Mallik · 4 months, 3 weeks ago

How many questions did you solve Harsh??Log in to reply

– Sayantan Saha · 4 months, 3 weeks ago

Our(West Bengal's) result has not come out.Log in to reply

– Sayantan Saha · 4 months, 3 weeks ago

I am from WB region. Result of our region has not come out.Log in to reply

– Vicky Vignesh · 5 months, 1 week ago

It was just 5 minutes after the exam was over LOLLog in to reply

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

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– Anik Mandal · 3 weeks, 3 days ago

It's easy.Which class for you? And did you write RMO?Log in to reply

Yep, That was easy try this. – Rohit Camfar · 3 weeks, 3 days ago

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– Anik Mandal · 3 weeks, 3 days ago

I am studying for boards now,So probably later..Log in to reply

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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– Harsh Shrivastava · 3 months, 3 weeks ago

I am selected from Chattisgarh region.Log in to reply

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

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– Wasif Jawad Hussain · 4 months, 1 week ago

51Log in to reply

– Wasif Jawad Hussain · 4 months, 1 week ago

Yours ???Log in to reply

– Mayank Jha · 1 month, 1 week ago

Wasif how much you scored in Inmo?Log in to reply

– Mayank Jha · 3 months, 3 weeks ago

Wasif what is the highest score in jharkhandLog in to reply

– Wasif Jawad Hussain · 4 months, 1 week ago

Yours?Log in to reply

– Mayank Jha · 4 months, 1 week ago

How much is the highest score in your city?Log in to reply

– Mayank Jha · 4 months, 1 week ago

What is the highest score you have heard till nowLog in to reply

– Wasif Jawad Hussain · 4 months, 1 week ago

Some other scores are 48,32...Log in to reply

– Mayank Jha · 4 months, 1 week ago

What is your kvpy expected scoreLog in to reply

– Wasif Jawad Hussain · 4 months, 1 week ago

50-55 as per old answer keyLog in to reply

– Mayank Jha · 4 months, 1 week ago

46.Can you tell how much chances I have to qualifyLog in to reply

– Wasif Jawad Hussain · 4 months, 1 week ago

By now i have heard 51(mine),34,41,...and waiting for others . do you know anyone with 60+ score ?Log in to reply

– Mayank Jha · 4 months, 1 week ago

You have not heard above 60 score till nowLog in to reply

– Wasif Jawad Hussain · 4 months, 1 week ago

What are the scores from jamshedpur ?Log in to reply

– Mayank Jha · 4 months, 1 week ago

Sorry I have no idea till now.Can you give me your mail addressLog in to reply

– Wasif Jawad Hussain · 4 months ago

What are the scores from jamshedpur ?????Log in to reply

– Mayank Jha · 4 months ago

When will the result of Rmo be declared?Log in to reply

– Mayank Jha · 4 months ago

Some scores are 15,25 many not received any emails.Log in to reply

– Mayank Jha · 4 months ago

Sorry wasif no information till now.my friends do not check their mail.Log in to reply

@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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– Anik Mandal · 5 months, 1 week ago

It is atleast one common language.Log in to reply

– Calvin Lin Staff · 5 months ago

Thanks. I've edited the question accordingly.Log in to reply

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

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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 agoLog in to reply

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– Calvin Lin Staff · 5 months, 1 week ago

The Farey sequence relates to question 3.Log in to reply

– Sharky Kesa · 5 months, 1 week ago

Yeah, sorry, looked at the wrong question. Question 4 looks like Pigeonhole Principle.Log in to reply

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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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 agoLog in to reply

– Manas Verma · 5 months, 1 week ago

*a-b is also in bracket.sane with a-cLog in to reply

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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– Wasif Jawad Hussain · 5 months, 1 week ago

You are in which class and did you got dps ranchi as exam centre ?Log in to reply

– Mayank Jha · 5 months, 1 week ago

I am from Jamshedpur,in class 11Log in to reply

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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– Kaustubh Miglani · 5 months, 1 week ago

You cannot apply AM GM Its not given that a-b,b-c,c-a are +veLog in to reply

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– Kaustubh Miglani · 5 months, 1 week ago

Yeah you are rightLog in to reply

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

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– Abhishek Alva · 5 months, 1 week ago

can u put brackets pleaseLog in to reply

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

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– Svatejas Shivakumar · 5 months, 1 week ago

Can you post the solution? Also when does equality occur?Log in to reply

– Kaustubh Miglani · 5 months, 1 week ago

Which questions did u do?Log in to reply

– Svatejas Shivakumar · 5 months, 1 week ago

Attempted all. Got 1,5,6 completely. Don't know the answer of 4. Did some progress in 1 and 2.Log in to reply

– Kaustubh Miglani · 5 months, 1 week ago

I got 3 questions fully correct and 1 question 25% corrrectLog in to reply

– Kaustubh Miglani · 5 months, 1 week ago

Dont u think q-1 is halwa? NCERT Question!!!Log in to reply

Really!! Though it was very easy but don't think it could be an NCERT question.– Saurabh Mallik · 5 months, 1 week agoLog in to reply

– Svatejas Shivakumar · 5 months, 1 week ago

Yes even 5 was very easy.Log in to reply

– Saurabh Mallik · 5 months, 1 week ago

Can you please post the complete solution of Q.5??Log in to reply

– Kushagra Sahni · 5 months, 1 week ago

Yes very easy everyone was able to solve it. Did you do Q.3?Log in to reply

– Kaustubh Miglani · 5 months, 1 week ago

Yeah did it Though I spent 55 mins on this oneLog in to reply

– Kushagra Sahni · 5 months, 1 week ago

Ok it took me also around 30-40 mins. How was RMO?Log in to reply

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

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– Kaustubh Miglani · 5 months, 1 week ago

It is better to evaluate it to be a+b+cLog in to reply

– Cdsc Dsc · 5 months, 1 week ago

It is better if u factorise given expression It is a+b+cLog in to reply

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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