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RMO-2015 Karnataka region

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.

  1. 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 \(ABC\).

  2. 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.

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

  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?

  5. 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 \(\Gamma\).

  6. 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!!

Note by Svatejas Shivakumar
1 year ago

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Answer 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 ago

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@Shivam Jadhav There is also two more solutions when a=b, (1,1,-3) and (-1,-1,3) Svatejas Shivakumar · 1 year ago

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@Shivam Jadhav Two more solutions! (4,4,3) and (-4,-4,-3) Ranjana Kasangeri · 1 year ago

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@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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@Svatejas Shivakumar Thanks !!! Naitik Sanghavi · 12 months ago

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@Svatejas Shivakumar Thanks but can you please search key for RMO telangana region. I am too busy ok. THANKS!!! Easha Manideep D · 12 months ago

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@Svatejas Shivakumar thanks Ganesh Ayyappan · 12 months ago

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@Svatejas Shivakumar Thanx :D Kush Singhal · 12 months ago

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@Kush Singhal How many did you solve? Do you have any idea what is the cut-off and when the results will be declared? Svatejas Shivakumar · 12 months ago

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@Svatejas Shivakumar I think I got none :/ Kush Singhal · 12 months ago

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

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@Naitik Sanghavi Can you send the link please? Svatejas Shivakumar · 1 year ago

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@Svatejas Shivakumar pccp.resonance.ac.in Naitik Sanghavi · 1 year ago

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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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@Easha Manideep D hi ... i had the same idea .. and proceeded in the same direction for Q6 ... i got 72 such integers ... is my answer right??

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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@Svatejas Shivakumar 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!! Naitik Sanghavi · 1 year ago

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@Naitik Sanghavi 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. Svatejas Shivakumar · 1 year ago

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@Svatejas Shivakumar So this case can be taken separately!and then we can solve ?? Naitik Sanghavi · 1 year ago

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@Naitik Sanghavi 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. Svatejas Shivakumar · 1 year ago

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@Svatejas Shivakumar Yes, even I realized just after writing this comment, hard luck. Anoushka Gupta · 1 year ago

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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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@Kush Singhal Well, may be its 72! Please add the list of numbers if you can! Ranjana Kasangeri · 1 year ago

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@Ranjana Kasangeri I got it wrong :( It's 72... Kush Singhal · 12 months ago

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@Kush Singhal How? Shashank Rammoorthy · 1 year ago

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@Shashank Rammoorthy Count the number of integers greater than a square above \(63^2\) and which is strictly lesser than the next square by 125... Kush Singhal · 1 year ago

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

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Comment deleted 12 months ago

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@Svatejas Shivakumar But 40 c 3 =9880 Adarsh Kumar · 1 year ago

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@Adarsh Kumar In 4th ,my answer is 46080 Naitik Sanghavi · 1 year ago

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Comment deleted Dec 07, 2015

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@Svatejas Shivakumar But your number is too large.Just look at your number once. Adarsh Kumar · 1 year ago

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Comment deleted 11 months ago

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@Svatejas Shivakumar I solved 3 perfectly but have a doubt in the combi one,if that is correct then 4. Adarsh Kumar · 1 year ago

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@Adarsh Kumar Did you also have the same paper? Svatejas Shivakumar · 1 year ago

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@Adarsh Kumar You solved 3 perfectly ,you will surely appear for INMO!0 Naitik Sanghavi · 1 year ago

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@Svatejas Shivakumar How many did you solved? Harsh Shrivastava · 1 year ago

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@Harsh Shrivastava Completely left question no.1 and did question no.5 with less than 5 minutes left :( Svatejas Shivakumar · 1 year ago

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@Harsh Shrivastava solved 3 not sure about question no.4 Svatejas Shivakumar · 1 year ago

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@Svatejas Shivakumar Which 3? Harsh Shrivastava · 1 year ago

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@Harsh Shrivastava What do you think would be the cutoff for this paper? Svatejas Shivakumar · 1 year ago

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@Svatejas Shivakumar I think minimum would be 34-40,and maximum 45-51! Naitik Sanghavi · 1 year ago

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@Naitik Sanghavi So is 34 like 2 questions fully right? Shashank Rammoorthy · 1 year ago

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@Naitik Sanghavi How many did you solve? Svatejas Shivakumar · 1 year ago

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@Svatejas Shivakumar In 3,i left one case where a=b! Naitik Sanghavi · 1 year ago

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@Svatejas Shivakumar I solved 3,4,6 3rd is right!(One case left) 6th is half (or less than that)Wrong! 4th don't know Naitik Sanghavi · 1 year ago

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@Naitik Sanghavi You too had the same paper! Svatejas Shivakumar · 1 year ago

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@Svatejas Shivakumar Each region had very similar paper!! Naitik Sanghavi · 1 year ago

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@Svatejas Shivakumar Yes but I'm from Gujarat Naitik Sanghavi · 1 year ago

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@Harsh Shrivastava How about you? Svatejas Shivakumar · 1 year ago

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@Harsh Shrivastava 2,3,6 Svatejas Shivakumar · 1 year ago

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