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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, 4 months 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 $(a,b,c)=(2,2,0),(-2,-2,0).$. · 1 year, 4 months ago

There is also two more solutions when a=b, (1,1,-3) and (-1,-1,3) · 1 year, 4 months ago

Two more solutions! (4,4,3) and (-4,-4,-3) · 1 year, 4 months ago

@naitik sanghavi @Ranjana Kasangeri @Ganesh Ayyappan @Kush Singhal @easha manideep d Official answer key has been declared click here. · 1 year, 4 months ago

Thanks !!! · 1 year, 4 months ago

Thanks but can you please search key for RMO telangana region. I am too busy ok. THANKS!!! · 1 year, 4 months ago

thanks · 1 year, 4 months ago

Thanx :D · 1 year, 4 months ago

How many did you solve? Do you have any idea what is the cut-off and when the results will be declared? · 1 year, 4 months ago

I think I got none :/ · 1 year, 4 months ago

Check Answer key by Resonance !! · 1 year, 4 months ago

Can you send the link please? · 1 year, 4 months ago

pccp.resonance.ac.in · 1 year, 4 months ago

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!!!! · 1 year, 4 months ago

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 .... · 1 year, 4 months ago

Oh by the way results came out. Go to isi Bangalore website and check out · 1 year, 4 months ago

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 . · 1 year, 4 months ago

How many questions do u need to get right to get qualify? · 1 year, 4 months ago

Answer to 4th one is 40C3 - 20.34 - 40.36 · 1 year, 4 months ago

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. · 1 year, 4 months ago

Comment deleted Dec 11, 2015

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!! · 1 year, 4 months 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. · 1 year, 4 months ago

So this case can be taken separately!and then we can solve ?? · 1 year, 4 months 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. · 1 year, 4 months ago

Yes, even I realized just after writing this comment, hard luck. · 1 year, 4 months ago

When will the results be out? · 1 year, 4 months ago

The 6th question's answer is 73 I think... · 1 year, 4 months ago

Well, may be its 72! Please add the list of numbers if you can! · 1 year, 4 months ago

I got it wrong :( It's 72... · 1 year, 4 months ago

How? · 1 year, 4 months ago

Count the number of integers greater than a square above $$63^2$$ and which is strictly lesser than the next square by 125... · 1 year, 4 months ago

Comment deleted Dec 11, 2015

But 40 c 3 =9880 · 1 year, 4 months ago

In 4th ,my answer is 46080 · 1 year, 4 months ago

Comment deleted Dec 07, 2015

But your number is too large.Just look at your number once. · 1 year, 4 months ago

Comment deleted Jan 13, 2016

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

Did you also have the same paper? · 1 year, 4 months ago

You solved 3 perfectly ,you will surely appear for INMO!0 · 1 year, 4 months ago

How many did you solved? · 1 year, 4 months ago

Completely left question no.1 and did question no.5 with less than 5 minutes left :( · 1 year, 4 months ago

solved 3 not sure about question no.4 · 1 year, 4 months ago

Which 3? · 1 year, 4 months ago

What do you think would be the cutoff for this paper? · 1 year, 4 months ago

I think minimum would be 34-40,and maximum 45-51! · 1 year, 4 months ago

So is 34 like 2 questions fully right? · 1 year, 4 months ago

yes · 1 year, 4 months ago

How many did you solve? · 1 year, 4 months ago

In 3,i left one case where a=b! · 1 year, 4 months ago

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

You too had the same paper! · 1 year, 4 months ago

Each region had very similar paper!! · 1 year, 4 months ago

Yes but I'm from Gujarat · 1 year, 4 months ago

How about you? · 1 year, 4 months ago