Time: 3 hours

Instructions:

\(\bullet\) Calculators (in any form) and protractors are not allowed.

\(\bullet\) Rulers and compasses are allowed.

\(\bullet\) Answer all the question.

\(\bullet\) All questions carry equal marks. Maximum marks: 102.

\(\bullet\) Answer to each question should start on a new page. Clearly indicate the question number.

Let \(ABC\) be a triangle. Let \(B'\) denote the reflection of \(b\) in the internal angle bisector \(l\) of \(\angle A\).Show that the circumcentre of the triangle \(CB'I\) lies on the line \(l\) where \(I\) is the incentre of \(ABC\).

Let \(P(x)=x^{2}+ax+b\) be a quadratic polynomial where \(a\) is real and \(b \neq 2\), is rational. Suppose \(P(0)^{2},P(1)^{2},P(2)^{2}\) are integers, prove that \(a\) and \(b\) are integers.

Find all integers \(a,b,c\) such that \(a^{2}=bc+4\) and \(b^{2}=ca+4\).

Suppose \(40\) objects are placed along a circle at equal distances. In how many ways can \(3\) objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?

Two circles \(\Gamma\) and \(\Sigma\) intersect at two distinct points \(A\) and \(B\). A line through \(B\) intersects \(\Gamma\) and \(\Sigma\) again at \(C\) and \(D\), respectively. Suppose that \(CA=CD\). Show that the centre of \(\Sigma\) lies on \(\Gamma\).

How many integers \(m\) satisfy both the following properties:

(i) \(1 \le m \le 5000\); (ii) \([\sqrt{m}]=[\sqrt{m+125}]\)?

(Here \([x]\) denotes the largest integer not exceeding \(x\), for any real number \(x\).)

Enjoy!!

## Comments

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TopNewestAnswer to question 3.\[a^{2}-b^{2}=c(b-a)\] This implies \[a+b+c=0 ...or... a=b\]

Case 1\[a+b+c=0\] Solutions are \[(a,b,c)=(2,0,-2),(-2,0,2),(0,2,-2),(0,-2,2),(2,2,0),(2,-2,0),(-2,2,0),(-2,-2,0).\]Case 2\[a=b \] Therefore \[(a,b,c)=(2,2,0),(-2,-2,0).\]. – Shivam Jadhav · 1 year agoLog in to reply

– Svatejas Shivakumar · 1 year ago

There is also two more solutions when a=b, (1,1,-3) and (-1,-1,3)Log in to reply

– Ranjana Kasangeri · 1 year ago

Two more solutions! (4,4,3) and (-4,-4,-3)Log in to reply

@naitik sanghavi @Ranjana Kasangeri @Ganesh Ayyappan @Kush Singhal @easha manideep d Official answer key has been declared click here. – Svatejas Shivakumar · 12 months ago

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– Naitik Sanghavi · 12 months ago

Thanks !!!Log in to reply

– Easha Manideep D · 12 months ago

Thanks but can you please search key for RMO telangana region. I am too busy ok. THANKS!!!Log in to reply

– Ganesh Ayyappan · 12 months ago

thanksLog in to reply

– Kush Singhal · 12 months ago

Thanx :DLog in to reply

– Svatejas Shivakumar · 12 months ago

How many did you solve? Do you have any idea what is the cut-off and when the results will be declared?Log in to reply

– Kush Singhal · 12 months ago

I think I got none :/Log in to reply

Check Answer key by Resonance !! – Naitik Sanghavi · 1 year ago

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– Svatejas Shivakumar · 1 year ago

Can you send the link please?Log in to reply

– Naitik Sanghavi · 1 year ago

pccp.resonance.ac.inLog in to reply

In 6th one the difference between two consecutive squares must be greater than 125. Then you can find step of sqrt m becomes step of sqrt of m + 125 which the smaller square in which m, m+125 lies. I gave a hint.

ENJOY!!!! – Easha Manideep D · 1 year ago

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aand cud sum1 post a link where i can get full detailed solution for this paper .... – Ganesh Ayyappan · 12 months ago

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Oh by the way results came out. Go to isi Bangalore website and check out – Kush Singhal · 11 months, 3 weeks ago

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For 3rd break into cases like @Shivam Jadhav did. But he forgot some cases, the two cases gets 8 and 6 subcases more for a+b+c=0 and a=b respectively . – Saarthak Marathe · 1 year ago

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How many questions do u need to get right to get qualify? – Tarush Govil · 1 year ago

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Answer to 4th one is 40C3 - 20.34 - 40.36 – Easha Manideep D · 1 year ago

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In Q 6, according to me, total no. of values of m are 8. 63^2 to 70^2. Verified. In Q3, even i have written permutations of (2,-2,0), total 6. But it's wrong. Someone pls do Q1 and 5. I wasted a lot of time on 1, have done something but couldn't conclude my answer. Pls do those 2 questions. – Anoushka Gupta · 1 year ago

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– Naitik Sanghavi · 1 year ago

My solution for 4th!(Tell mewhat is wrong in my solution)-(I'm writing in short) There are 40 ways to select 1st object! There are 36 ways to select 2nd! And ,32 ways to select the 3rd! In total there are 40×36×32=46080 ways!!Log in to reply

– Svatejas Shivakumar · 1 year ago

There is a mistake. If the second object is two places away from the first object, you count the restricted object twice. Sorry. Good luck for getting selected.Log in to reply

– Naitik Sanghavi · 1 year ago

So this case can be taken separately!and then we can solve ??Log in to reply

– Svatejas Shivakumar · 1 year ago

Also I don't think you can multiply 40 and 36 and so on because there will be several cases in which you choose the Same object. I just realized that repetitions are not allowed as they are placed around a circle. Hard luck.Log in to reply

– Anoushka Gupta · 1 year ago

Yes, even I realized just after writing this comment, hard luck.Log in to reply

When will the results be out? – Shashank Rammoorthy · 1 year ago

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The 6th question's answer is 73 I think... – Kush Singhal · 1 year ago

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– Ranjana Kasangeri · 1 year ago

Well, may be its 72! Please add the list of numbers if you can!Log in to reply

– Kush Singhal · 12 months ago

I got it wrong :( It's 72...Log in to reply

– Shashank Rammoorthy · 1 year ago

How?Log in to reply

– Kush Singhal · 1 year ago

Count the number of integers greater than a square above \(63^2\) and which is strictly lesser than the next square by 125...Log in to reply

Please tell your answers to 3,4,6 questions!! – Naitik Sanghavi · 1 year ago

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– Adarsh Kumar · 1 year ago

But 40 c 3 =9880Log in to reply

– Naitik Sanghavi · 1 year ago

In 4th ,my answer is 46080Log in to reply

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– Adarsh Kumar · 1 year ago

But your number is too large.Just look at your number once.Log in to reply

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– Adarsh Kumar · 1 year ago

I solved 3 perfectly but have a doubt in the combi one,if that is correct then 4.Log in to reply

– Svatejas Shivakumar · 1 year ago

Did you also have the same paper?Log in to reply

– Naitik Sanghavi · 1 year ago

You solved 3 perfectly ,you will surely appear for INMO!0Log in to reply

@Nihar Mahajan @Saarthak Marathe @Shrihari B @Harsh Shrivastava @Dev Sharma @Swapnil Das – Svatejas Shivakumar · 1 year ago

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– Harsh Shrivastava · 1 year ago

How many did you solved?Log in to reply

– Svatejas Shivakumar · 1 year ago

Completely left question no.1 and did question no.5 with less than 5 minutes left :(Log in to reply

– Svatejas Shivakumar · 1 year ago

solved 3 not sure about question no.4Log in to reply

– Harsh Shrivastava · 1 year ago

Which 3?Log in to reply

– Svatejas Shivakumar · 1 year ago

What do you think would be the cutoff for this paper?Log in to reply

– Naitik Sanghavi · 1 year ago

I think minimum would be 34-40,and maximum 45-51!Log in to reply

– Shashank Rammoorthy · 1 year ago

So is 34 like 2 questions fully right?Log in to reply

– Svatejas Shivakumar · 1 year ago

yesLog in to reply

– Svatejas Shivakumar · 1 year ago

How many did you solve?Log in to reply

– Naitik Sanghavi · 1 year ago

In 3,i left one case where a=b!Log in to reply

– Naitik Sanghavi · 1 year ago

I solved 3,4,6 3rd is right!(One case left) 6th is half (or less than that)Wrong! 4th don't knowLog in to reply

– Svatejas Shivakumar · 1 year ago

You too had the same paper!Log in to reply

– Naitik Sanghavi · 1 year ago

Each region had very similar paper!!Log in to reply

– Naitik Sanghavi · 1 year ago

Yes but I'm from GujaratLog in to reply

– Svatejas Shivakumar · 1 year ago

How about you?Log in to reply

– Svatejas Shivakumar · 1 year ago

2,3,6Log in to reply