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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TopNewest@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 · 2 days, 11 hours ago

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– Anik Mandal · 2 days, 10 hours ago

It is atleast one common language.Log in to reply

– Calvin Lin Staff · 2 days, 3 hours ago

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

Anyone with 6 th done – Aakash Khandelwal · 3 days, 7 hours ago

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Grasp here and here – Vicky Vignesh · 3 days ago

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20 is the answer for the 4th one – Mayank Jha · 4 days, 10 hours 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 · 4 days, 23 hours ago

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solutions are now uploaded on resonance. – Svatejas Shivakumar · 5 days, 10 hours ago

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– Vicky Vignesh · 4 days, 23 hours ago

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

what is the answer of the 4 th one – Abhishek Alva · 5 days, 21 hours 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 days, 21 hours ago

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Has anyone perfectly solved Q.4??– Saurabh Mallik · 6 days, 7 hours agoLog in to reply

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– Calvin Lin Staff · 2 days, 22 hours ago

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

– Sharky Kesa · 2 days, 15 hours 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 · 6 days, 7 hours 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 · 6 days, 17 hours 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 · 6 days, 7 hours agoLog in to reply

– Manas Verma · 6 days, 17 hours 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 · 6 days, 17 hours ago

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@Svatejas Shivakumar Could you post the solution to problem 6? – Milind Prabhu · 6 days, 23 hours ago

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

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– Wasif Jawad Hussain · 5 days, 21 hours ago

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

– Mayank Jha · 4 days, 10 hours ago

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

Same paper was for chhattisgarh also! – Manas Verma · 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 · 1 week ago

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– Kaustubh Miglani · 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 · 1 week ago

Yeah you are rightLog in to reply

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

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– Abhishek Alva · 1 week ago

can u put brackets pleaseLog in to reply

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

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– Svatejas Shivakumar · 1 week ago

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

– Kaustubh Miglani · 1 week ago

Which questions did u do?Log in to reply

– Svatejas Shivakumar · 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 · 1 week ago

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

– Kaustubh Miglani · 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 · 6 days, 7 hours agoLog in to reply

– Svatejas Shivakumar · 1 week ago

Yes even 5 was very easy.Log in to reply

– Saurabh Mallik · 6 days, 7 hours ago

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

– Kushagra Sahni · 1 week ago

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

– Kaustubh Miglani · 6 days, 11 hours ago

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

– Kushagra Sahni · 6 days, 8 hours ago

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

Both went good – Kaustubh Miglani · 4 days, 11 hours ago

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– Kaustubh Miglani · 1 week ago

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

– Cdsc Dsc · 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 · 6 days, 10 hours ago

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