So why aren't we considering negative values of \(tan\) because \(tan\) ranges from -infinity to +infinity.

I think that we should also include in the proof that the triangle is an acute angled triangle. (because in acute angled triangle, all angles will be less than \(90\) degrees and since we know that \(tan\) is positive in first quadrant \((0\) to \(90)\) degrees, we can directly eliminate the negative values of \(tan\) in the question.)

So, therefore I think you are not considering the negative values of \(\boxed{tan(B)}\).
–
Saurabh Mallik
·
8 months, 1 week ago

I think we have to mention in the solution that since the orhtocentre of the triangle lies inside the triangle, it's an acute angled triangle, and therefore we are taking only the positive values of \(tan(B)\).

Shourya Pandey how many questions did you solve in INMO?Which books did you study for RMO and INMO??
–
Saurabh Mallik
·
8 months, 1 week ago

Synthetic solution to number 1: Since the triangle is isosceles, \(A,I,H\) are collinear, where \(I,H\) are the incenter and orthocenter respectively. Let \(AH\cap BC, \odot (ABC)=D,E\). It is well known that \(DE=DH\), thus \(H\) lies on the the incircle implies \(DE=DH=2ID\). Using an incenter property from triangles - incenter, we know

Solution to number 6: The sequence is arithemtic so we can let \(a_n=a_1+(n-1)d\). Note that \(m=a_i\) for some \(i\) \(\iff \) \(\displaystyle \frac {m-a_1}{d}\in \mathbb{Z}\). Since \(a_{p+1}=a_1+pd, a_{q+1}=a_1+qd\), the following is true:
\[\frac {a_1^2-a_1}{d}\in \mathbb{Z}\\ \frac {a_1^2-a_1}{d}+2a_1p+p^2d\in \mathbb{Z}\\ \frac {a_1^2-a_1}{d}+2a_1p+p^2d\in \mathbb{Z}\]

It is not difficult to show that \(a_1,d\in \mathbb{Q}\), thus we can assume \(\displaystyle a_1=\frac {x_1}{y_1}, d=\frac {x_2}{y_2}\) with \(x_1,y_1,x_2,y_2\in \mathbb{Z}\) and \((x_1,y_1)=1, (x_2,y_2)=1\). It suffices to show that \(y_1=y_2=\pm 1\).

Plugging these into the first expression gives \(\frac {(x_1^2-x_1y_1)y_2}{y_1^2x_2}\in \mathbb{Z}\iff y_1^2x_2|(x_1^2-x_1y_1)y_2\). We can simplify this further by considering the relatively prime condition. Note that \((y_1,x_1)=1\implies (y_1,x_1-y_1)=1\), thus \((y_1^2, x_1^2-x_1y_1)=1\). Hence \(\displaystyle y_1^2|y_2 (\star)\).

Applying the same substitution for the second equation gives \(y_1y_2|2px_1y_2+p^2x_2y_1\implies y_2|p^2x_2y_1\implies\) \(\displaystyle y_2|p^2y_1\). From \((\star\) we can let \(y_2=y_1^2k\) for some integer \(k\), thus \(y_2|p^2y_1\implies y_1^2k|p^2y_1\implies y_1k|p^2\). Similarly, the analogous operation on the third equation gives \(y_1k|q^2\). This means \(y_1k\) is a common divisor of \(p^2,q^2\). Since \((p^2,q^2)=1\), \(|y_1k|=1\implies |y_1|=1, |y_2|=1\), which is what we want to prove.
–
Xuming Liang
·
8 months, 1 week ago

Log in to reply

@Xuming Liang
–
I could reach to the equation (*) but then beated around the Bush after that how much will i get on this question then?
–
Samarth Agarwal
·
8 months, 1 week ago

Log in to reply

@Samarth Agarwal
–
You can get 3/4 of the total marks of the question for your solution. If a question is of 16.6 marks then i think you should get around 12 marks.
–
Priyanshu Mishra
·
8 months, 1 week ago

@Xuming Liang
–
I could only prove a and d are rational. How much should I expect ?( I did mention that those 3 expressions were integers.)
–
Rohit Kumar
·
8 months, 1 week ago

Log in to reply

@Xuming Liang
–
I did the same. I really don't think there is any other solution to it. Nice write up, by the way.
–
Shourya Pandey
·
8 months, 1 week ago

Log in to reply

@Xuming Liang
–
Nice solution. Sequence problems are very fond for the problem posers as it requires many topics.
–
Priyanshu Mishra
·
8 months, 1 week ago

I am Anay Karnik from Mumbai, just received my scorecard, got 51.
Let's make a list to determine the cut-off.

Anay Karnik (Mumbai) - 51

By the way, the question paper uploaded is actually mine (I know from the wrinkles on the bottom of the paper) I was the first one to reach home, and scan my paper and upload it on AOPS. The person who uploaded it here has copied mine from there!
–
Anay Karnik
·
7 months, 3 weeks ago

@Raushan Sharma
–
I just checked in with the list in AOPS where lot of people have posted their marks and some people are predicting 65 cut-off.
–
Anay Karnik
·
7 months, 3 weeks ago

@Abhishek Bakshi
–
No 55 is too low for a cutoff this year. I agree with Anay Karnik. The cutoff will be 63-65. And hi Anay this is me Shrihari :)
–
Shrihari B
·
7 months, 2 weeks ago

Log in to reply

@Abhishek Bakshi
–
No, I also saw that list in AOPS, yes Anay is right, cutoff might be around 65 this time
–
Raushan Sharma
·
7 months, 2 weeks ago

Log in to reply

@Raushan Sharma
–
@Shrihari B, @Raushan Sharma: The cutoff really doesn't matter much to me... but even if the paper is easy, the probability that the cutoff will go above 60 is really low... well anyway I hope you guys get selected whatever be the cutoff... :-D
–
Abhishek Bakshi
·
7 months, 2 weeks ago

Log in to reply

@Anay Karnik
–
Yes I copied it from Aops. Don't remember who had posted it but I guess it must be you :P I had earlier mentioned in the note that I copied it from Aops as I didn't write INMO. Probably a moderator edited it. Anyway thanks for uploading the paper, had great fun solving or at least trying to solve the problems :)
–
Svatejas Shivakumar
·
7 months, 3 weeks ago

(i)By the well ordering principle the set of positive integral values of \(T^k(n)\) must have a least value(positive integral) say m. Let's assume that \(m \neq 1\).

If m is even:\[T(m)=\dfrac{m}{2}<m\]

If m is odd:\[T(T(m))=\dfrac{m+1}{2}<m\]

Both cases lead to a contradiction. Hence our assumption is wrong.
–
Svatejas Shivakumar
·
8 months, 1 week ago

A short proof to problem 4 is to consider 2017 very close points, say, the angle made by the minor arc formed by the farthest two points is less than \(\frac {2\pi}{n} \) . Complete the 2017 regular n-gons formed by these points. Clearly(one shouod explain this in the examination) no two n-gons have any points in common. Therefore, by the pigeonhole principle, at least one of these n-gons has no red point as its vertex.
–
Shourya Pandey
·
8 months, 1 week ago

Log in to reply

@Shourya Pandey
–
Hey, Shourya, tell me what's wrong in this proof. Let's label 2 adjacent red points to be \(A_1\) and \(A_2\). Now we divide the arc between the 2 point into an arbitrarily large number of equal parts, say, \(10^9\), and we mark points labelled as \(R_1\, R_2\) and so on to denote the boundary of the sectors. Now we draw a regular n-gon with \(A_1\) as one of the vertex. It may or may not contain any other red point. Now we rotate the n-gon so that the vertex shifts to \(R_1\). This n-gon also may have a red point on it. If it doesn't then this is our required polygon. If it does then we rotate it again to \(R_2\) and so on. We have only 2016 red points hence a maximum number of 2015 of the regular n-gons out of the \(10^9\) polygons so created will have red points as one of the vertices. Hence the others are our required polygons. Hence proved.
–
Abhishek Bakshi
·
8 months ago

Log in to reply

@Abhishek Bakshi
–
That's pretty much same to what Shourya did. I think it's correct.
–
Rohit Kumar
·
7 months, 3 weeks ago

Log in to reply

@Abhishek Bakshi
–
I am not sure about it. But I feel that you should have proved that the the n-gons are regular. See the arrangement of the points is not regular in the case you have mentioned. So getting a regular n-gon will be a very special case. You need to prove that the special case exists.
–
Aditya Kumar
·
7 months, 3 weeks ago

Log in to reply

@Aditya Kumar
–
Huh????? what are you talking about... you want an n-gon with blue vertices which are infinite in number in contrast to the red ones which are only 2016 in number. A regular n-gon with blue vertices is not a special case..
–
Abhishek Bakshi
·
7 months, 3 weeks ago

Log in to reply

@Abhishek Bakshi
–
The question is concerned about obtaining a regular n-gon. For that you have to prove that the sides are equal and all angles are equal. Now according to your explanation, the \(R_{n}\) distribution is irregular. Hence obtaining a regular n-gon becomes a special case.
–
Aditya Kumar
·
7 months, 3 weeks ago

Log in to reply

@Aditya Kumar
–
I think you have misunderstood my solution..I am not saying that a regular n-gon will be formed using the points of \(R_n\). I am saying that just take one point of \(R_n\)and draw a regular n-gon with it as one of the vertices. Such a polygon always exists and we have to just prove that out of the \(10^9\) polygons such formed, the 2016 red points don't lie on all of them which is quite obvious. I believe that you understand the sarcasm too...haha..
–
Abhishek Bakshi
·
7 months, 3 weeks ago

A lengthy proof to problem 4:
Let the circle given in the question be the unit circle centred at the origin, WLOG.
Each point has a unique polar coordinate \( (1, \theta) \), where \( 0 \le \theta <2\pi \). We will express all points (on this circle) only by its polar angle.
Let \(\theta_{i} , i= 1,2,3,...,2016 \) denote the 2016 red points on the circle. For convinience, denote \( \frac {\theta}{\pi} \) as \(b_i\)

LEMMA 1 :- There exists a number \(\alpha \), such that \( \alpha - b_i\) is irrational, for all \(i\).
(Try proving this. I'll post the proof later).

Now, we choose the points of the regular n-gon to be \( \alpha\pi , \alpha\pi + \frac {2\pi}{n} , ..., \alpha\pi + \frac {2(n-1)\pi}{n} \). Here \(\alpha \) may not lie in \([0, 2\pi) \), but it does not matter.

Suppose some vertex of this n-gon coincides with some red point, say \( \alpha\pi + \frac {2l\pi}{n} = \theta_i+ 2k\pi\), where \(l\) is one of\(0,1,2,...,n-1\) and \(k\) is an integer. Then rearranging it gives
\(\alpha - b_i= 2k - \frac {2l}{n} \), a contradiction to the lemma.

This solves the problem.
–
Shourya Pandey
·
8 months, 1 week ago

Log in to reply

Anyone who cleared INMO?
–
Mayank Jha
·
7 months ago

Log in to reply

@Mayank Jha
–
I did (got 73). You should join the discussion on AoPS, my username is TheOneYouWant there... looks like someone below was impersonating me :P
–
Shubham Jain
·
7 months ago

Log in to reply

@Shubham Jain
–
Thanks guys! A lot of people have been asking me the same question lately, thus I have written an article on it on my site w/ my friends: http://mathometer.weebly.com/preparing-for-inmo.html
The site is a mini success, garnering 12k views in the 3 weeks of its existence :)
–
Shubham Jain
·
6 months, 3 weeks ago

Log in to reply

@Shubham Jain
–
Congrats! Shubham on getting selected for INMO. Which standard you are currently in?? Can you tell/guide me on how did you prepare for INMO please??
–
Saurabh Mallik
·
7 months ago

Log in to reply

@Shubham Jain
–
Pls can you advise me on how to prepare effectively for INMO.
I am not in AOPS.
–
Mayank Jha
·
7 months ago

@Shourya Pandey
–
Do they award full marks ?(I heard from someone that they don't award you full marks even if your solution is correct )
–
Rohit Kumar
·
7 months, 3 weeks ago

Log in to reply

@Rohit Kumar
–
No. They do award full marks. If you are able to prove something crucial using coordinate geometry and you mention the geometrical interpretation of the coordinate bash you have done, then marks are awarded according to how crucial it is in the pure geometric solution. However, if you are just blindly doing algebra and reach some weird coordinates ( even though you are really close), you get a zero. Also, you lose all marks if you say " clearly .... simplifies to ... ", where it isn't very clear how the previous step would mean the next.

This is why it is often recommended to use coordinate only as a tool to prove small steps in the problems. Of course, it is different if you're very good at bashing.
–
Shourya Pandey
·
7 months, 3 weeks ago

Log in to reply

@Shourya Pandey
–
In the sixth question, I was able to prove that a and d were rational. How much should I expect ?
–
Rohit Kumar
·
7 months, 3 weeks ago

Log in to reply

Hey guys I would Like if everyone posted their expected marks here. That would help us to get a rough idea of the cutoff. Starting with me ... I am expecting 34 :( After u post your marks I will edit it in here and then delete your comments to avoid a lot of junk.
–
Shrihari B
·
8 months, 1 week ago

Log in to reply

@Shrihari B
–
A friend of mine got five and a half questions correct.
–
Racchit Jain
·
8 months, 1 week ago

(I)Let \(a=b=1\). Therefore \(c^3-2c+1=0\). Solving we get a solution as \(c=\dfrac{-1+\sqrt{5}}{2}\). Hence it is not necessary for \(a=b=c\)

(II)We may assume \(a=max(a,b,c)\) since \(a(a^3+b^3)=b(b^3+c^3)\) and \(a \ge b\), \(a^3+b^3 \le b^3+c^3\) or \(c \ge a\). This forces \(a=c\). Similarly, \(a(a^3+b^3)=c(c^3+a^3)\) or \(b=c\).

@Svatejas Shivakumar
–
The second part is NOT SYMMETRIC, but only CYCLIC. Therefore \(a \ge b \ge c\) and \(a \ge c \ge b\) are two different cases.
–
Shourya Pandey
·
8 months, 1 week ago

@Shourya Pandey
–
Hey Shourya I didn't take the second case i.e. a>=c>=b. How many marks will be deducted ???
–
Shrihari B
·
8 months, 1 week ago

Log in to reply

@Shourya Pandey
–
I assumed symmetric and the first inequality you wrote .how much should I get?
–
Aman Anand
·
8 months, 1 week ago

Log in to reply

@Aman Anand
–
You cannot assume the symmetric case as Shourya mentioned. I am not sure how much you would get but I guess 6-7. Hard luck.
–
Svatejas Shivakumar
·
8 months, 1 week ago

Log in to reply

@Svatejas Shivakumar
–
I solved 1,2,3 please see no. 3 which i posted above.I assumed a>=b>=c but not a>=c>=b in the second one and justified the first case of no. 2.how much should I get in no. 2.
–
Aman Anand
·
8 months, 1 week ago

@Svatejas Shivakumar
–
Yes, absolutely correct. Just I assumed a=b=2, and then c=\(\sqrt{5} - 1\) for the 1st one, and 2nd one same assumption WLOG, and then some manipulation.
–
Raushan Sharma
·
8 months, 1 week ago

Log in to reply

You know anyone from your region and how much he/she did?
–
Raushan Sharma
·
8 months, 1 week ago

Got the scorecard....could only make 35.... :(
–
Samarth Agarwal
·
7 months, 2 weeks ago

Log in to reply

I wanna ask is this correct for no. 3
T(2)=1
T(3)=4,=>T^(3)=1
T(4)=2,=>T^2(4)=1
1)We prove by induction.Assume for some 4 \leq m \geq 2n and some k,we can get T^k(m)=1.
Case 1) for m=2n+1
T(m)=2n+2;=>T^2(m)=n+1 < 2n
So we can get for some k, T^k(2n+1)=1
Similarly for case 2) .So this proves our assumption.
2) T^{k+2} (n)=1;for c{k+2} values.
Case 1)n=2m
T^{k+2} (2m)=T^{k+1} (m)=1;for c{k+1} values.
Case 2)n=2m-1
T^{k+2} (2m-1)=T^{k+1} (2m)=T^k (m)=1;for c{k} values.
This shows c{k+2}=c{k+1}+c{k}.
–
Aman Anand
·
8 months, 1 week ago

Log in to reply

4.

Isn't this one very simple? There are infinitely many points which are blue. There are also infinitely many points which are blue between two red points. There are only 2016 points, so definitely there are infinitely many regular n-gons which means that there are infinitely many regular n-gons with blue vertices.

@Svatejas Shivakumar
–
Yes, this question I couldn't understand what it meant, or what I needed to show, coz there are infinitely many points on the circle, so... what does this mean!!
–
Raushan Sharma
·
8 months, 1 week ago

@Akash Deep
–
I think the papers were comparable. Although the paper this year was slightly difficult on an average, but the 5th question this year wasn't impossible in the time-limit ( unlike last year's).
–
Shourya Pandey
·
8 months, 1 week ago

Log in to reply

@Shourya Pandey
–
Are you sure ? Because 1st, 2nd and 4th were easy.(I might be wrong)
How many did you solve ?
–
Rohit Kumar
·
8 months, 1 week ago

@Samarth Agarwal
–
Construct a regular polygon with one of its vertices fixed at a red point.
Now measure the positive anticlockwise angles between all vertice and all red point.
Rotate the polygon by half of the smallest of these angles.
Now every vertex initially on a blue point must land on a blue point, as the rotation was by half of the smallest angle.
Every vertex on a red point must land on a blue point, as the polygon was rotated.

If a vertex is already on a red point, measure the angle from adjacent red points.

@Rohit Kumar
–
I m weak at combinotorics but this seems correct. ...btw how many did u got correct n in which class r u?
–
Samarth Agarwal
·
8 months, 1 week ago

Log in to reply

@Samarth Agarwal
–
I solved 1st , 2nd, 4th completely, solved part 1 of 3rd, and I couldn't complete 6th.
I am in 11th.
–
Rohit Kumar
·
8 months, 1 week ago

Log in to reply

@Rohit Kumar
–
Can anybody please explain me question no. \(3\) properly?

Rohit you belong to which region? Which books did you study for RMO and INMO??
–
Saurabh Mallik
·
8 months, 1 week ago

@Raushan Sharma
–
That's exactly the solution I wrote in the exam! Don't tell it's incomprehensible!
–
Rohit Kumar
·
8 months, 1 week ago

Log in to reply

@Rohit Kumar
–
No, it isn't u will get 17 definitely. I understood now, but for that I had to draw figures and all, BTW, nice solution
–
Raushan Sharma
·
8 months, 1 week ago

Log in to reply

@Shourya Pandey
–
Would you guide me to prepare for Rmo and INMO?Can you give me your mail address?
–
Mayank Jha
·
8 months, 1 week ago

@Samarth Agarwal
–
I have also solved 2 and a half, but I think that's not enough for selection, but yeah, we can expect being in the merit list
–
Raushan Sharma
·
8 months, 1 week ago

Log in to reply

@Shourya Pandey
–
i did 1,
2 (in 2nd i just showed for the first case that all 3 of them cannot be distinct , means any 2 are equal , after taking this i showed that if a=b=x and x and c satisfy x^3 +c^3 - 2c^2x = 0 then there is no need for x = c)whereas the 2nd part of 2nd question i did correctly.
in 3rd i did first part correctly and if s(k) denotes the set containing elements for which t^k(n) in ,2nd part i showed that s(k) is a subset of s(k+2) and showed that from every element of s(k+1) we can generate a corresponding element of s(k+2), then i showed that if there is any element in s(k+2) it is either of s(k) or generated from s(k+1)
4- did not attempt
5- wrote that R*S stuff and left it.
6. - just wrote that for an a.p to have all elements integal , c.d and first term must be integral with a useless proof of this.
(please let me know if i have any chances and how much marks can i get)
–
Akash Deep
·
8 months, 1 week ago

Log in to reply

@Akash Deep
–
Many people are getting only 2 to 3 questions correct so may be cutoff ho down
–
Samarth Agarwal
·
8 months, 1 week ago

Log in to reply

Does anyone knows what will be the cutoff this year
–
Samarth Agarwal
·
8 months, 1 week ago

@Harsh Shrivastava
–
I thought u are in class 11....but if u have done 2 then u may get inmo merit certificate and direct eligibility for inmo next year
–
Samarth Agarwal
·
8 months, 1 week ago

@Harsh Shrivastava
–
Do u know the solution top fifth one? ......using rs=area I reached somewhere and I needed only to prove that bd^2=area of abc.....not getting after that
–
Samarth Agarwal
·
8 months, 1 week ago

@Svatejas Shivakumar
–
I also got the first part fully, but in the second part I tried many things, but didn't land up with the result. What do u think is the partition of marks in Q.3??
–
Raushan Sharma
·
8 months, 1 week ago

Log in to reply

@Raushan Sharma
–
Since i am using brilliant on phone its difficult to use latex once my pc is repaired i would write a sol. To 3rd one ....its very interesting
–
Samarth Agarwal
·
8 months, 1 week ago

@Svatejas Shivakumar
–
I would soon write a complete sol. Of ques. 3.... the second part involves cases taking a no. In the set to be once even and then odd... (I would post complete solution soon)
–
Samarth Agarwal
·
8 months, 1 week ago

Log in to reply

@Samarth Agarwal
–
if the length of the cevian is <p> then the ratio it divides the hypotenuse is a+p/c+p.where c,a are its sides
–
Deekshith Kanagala
·
8 months, 1 week ago

Log in to reply

@Samarth Agarwal
–
Yes, I also got to the same to prove BD^2 = Area, then I was trying with Stewart's theorem, and also dropped perpendiculars from A and C on BD, and tried to find the value of BD^2 in two different ways, and equate, but the time was up, and it was incomplete LOL. I think this problem was quite manipulative.
–
Raushan Sharma
·
8 months, 1 week ago

@Raushan Sharma
–
10 is very much for 3 statements....may be 5?......but i just hope u r right
–
Samarth Agarwal
·
8 months, 1 week ago

Log in to reply

@Samarth Agarwal
–
No, no, actually one of my fellow mates Vishal Raj solved it fully using Stewart's theorem only, and his strategy was also the same as writing BD^2 in two different ways, and then manipulating. So, we can expect 10 marks I hope
–
Raushan Sharma
·
8 months, 1 week ago

@Aditya Kumar
–
I am able to see it clearly. If you are still not able to see the problems, I will type the problems here.
–
Svatejas Shivakumar
·
8 months, 1 week ago

@Deekshith Kanagala
–
@deekshith kanagala: how much are you expecting... I think this year's cutoff will be higher than that of last year... it should be around 55.. those who get selected would have solved the 1st, 2nd and 4th problems fully and 1st part of the 3rd problem...
–
Abhishek Bakshi
·
7 months, 3 weeks ago

@Deekshith Kanagala
–
I also did 1st and 3rd full and did 5th,6th partially......is that enough for qualifying?...has anybody received the scorecards?
–
Samarth Agarwal
·
7 months, 3 weeks ago

@Deekshith Kanagala
–
@deekshith kanagala: nice score... you are quite accurate as you got the marks for 2 questions.... but i don't think the cutoff will go so low... btw I am in 12th standard and i gave inmo last 2 years... unfortunately i missed the cutoff last year by 6 marks, so i have a little knowledge about what the cutoff could be...
–
Abhishek Bakshi
·
7 months, 2 weeks ago

## Comments

Sort by:

TopNewestAnswer to question number 1:Let the foot of the perpendicular from \(A\) to \(BC\) be \(M\).

Let \(H,I\) be the orthocentre and incentre respectively.

(I)Since in an isosceles triangle orthocentre and incentre are collinear .

Proof of

statement I:\(\triangle(ABM)\) is congruent to \(\triangle(ACM)\) (By \(RHS\) test ) \(AB=AC,AM=AM,\angle(AMB)=\angle(AMC)\)

Therefore \(M\) is a midpoint of \(BC\). Let \(J\) be the intersection of incircle with \(BC\).

Therefore \(\angle(IJC)=90°\). \(2BJ=AB+BC-AC,2CJ=BC+AC-AB\)

But, \(AB=AC\)

Therefore \(BJ=CJ,BJ+CJ=BC\) .

This implies \(J\) is the midpoint of \(BC\) , but we proved that \(M\) is the midpoint of \(BC\) .

Therefore \(J,M\) must coincide.

This proves that \(I\) lies on \(AM\).

Therefore \(A,M,H,I\) are colinear.

Since \(AB=AC\) ,\(\angle(B)=\angle(C)\) Now \(\boxed{tan(\frac{B}{2})=\frac{IM}{BM}}.......(1)\) ) \(\angle(HBC=(90-C)=(90-B)\)

Therefore \(tan(90-B)=\frac{HM}{BM}\)

But \(2IM=HM\) since \(H,I,M\) are collinear and also lie on a circle with \(I\) as the centre .

hence \(\boxed{Cot(B)=\frac{2IM}{BM}}.......(2)\)

Dividing \((1)\) by \((2)\) We get \(\frac{1}{2}=tan(B)tan(\frac{B}{2})\)

After solving this we get, \(tan(B)=\frac{\sqrt{5}}{2}\)

But \((90-\frac{A}{2})=B\) Therefore \(tan(\frac{A}{2})=\frac{2}{\sqrt{5}}\)

This implies \(sin(\frac{A}{2})=\frac{2}{3}......(3)\)

In triangle \(ABM\) \(sin(A/2)=BM/AB=BC/2AB....(4)\)

Therefore using \((3),(4)\) we get

\[\frac{AB}{BC}=\frac{3}{4}\] – Shivam Jadhav · 8 months, 1 week ago

Log in to reply

Shivam Jadhav I have a doubt.After solving \(tan(B)tan(\frac{B}{2})=\frac{1}{2}\), I'm getting the following values:-

\(tan(\frac{B}{2})=\frac{1}{\sqrt{5}}, -\frac{1}{\sqrt{5}}\)

\(tan(B)=\frac{\sqrt{5}}{2}, -\frac{\sqrt{5}}{2}\)

So why aren't we considering negative values of \(tan\) because \(tan\) ranges from

-infinity to +infinity.I think that we should also include in the proof that the triangle is an acute angled triangle. (because in acute angled triangle, all angles will be less than \(90\) degrees and since we know that \(tan\) is positive in first quadrant \((0\) to \(90)\) degrees, we can directly eliminate the negative values of \(tan\) in the question.)So, therefore I think you are not considering the negative values of \(\boxed{tan(B)}\).– Saurabh Mallik · 8 months, 1 week agoLog in to reply

– Shivam Jadhav · 8 months, 1 week ago

Ratio of two sides can't be negative .Log in to reply

And we know that value of \(tan\) is negative in the second quadrant (between \(90\) to \(180\) degrees). – Saurabh Mallik · 8 months, 1 week ago

Log in to reply

– Shourya Pandey · 8 months, 1 week ago

Indeed. It should be mentioned in the solution, at least.Log in to reply

I think we have to mention in the solution that since the orhtocentre of the triangle lies inside the triangle, it's an acute angled triangle, and therefore we are taking only the positive values of \(tan(B)\).Shourya Pandey how many questions did you solve in INMO?Which books did you study for RMO and INMO??– Saurabh Mallik · 8 months, 1 week agoLog in to reply

– Shourya Pandey · 7 months, 3 weeks ago

I didn't appear for the INMO.Log in to reply

Synthetic solution to number 1: Since the triangle is isosceles, \(A,I,H\) are collinear, where \(I,H\) are the incenter and orthocenter respectively. Let \(AH\cap BC, \odot (ABC)=D,E\). It is well known that \(DE=DH\), thus \(H\) lies on the the incircle implies \(DE=DH=2ID\). Using an incenter property from triangles - incenter, we know\[\frac {AI}{ID}=\frac {EI}{DE}=\frac {ID+DE}{DE}=\frac {3}{2}\].

Hence by the angle bisector theorem, \[\frac {AB}{BC}=\frac {AB}{2BD}=\frac {AI}{2ID}=\boxed {\frac {3}{4}}\] – Xuming Liang · 8 months, 1 week ago

Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

This is the shortest and easiest way to solve the question and get 17/17......+1 :)Log in to reply

5TH QUESTION

Log in to reply

Solution to number 6: The sequence is arithemtic so we can let \(a_n=a_1+(n-1)d\). Note that \(m=a_i\) for some \(i\) \(\iff \) \(\displaystyle \frac {m-a_1}{d}\in \mathbb{Z}\). Since \(a_{p+1}=a_1+pd, a_{q+1}=a_1+qd\), the following is true: \[\frac {a_1^2-a_1}{d}\in \mathbb{Z}\\ \frac {a_1^2-a_1}{d}+2a_1p+p^2d\in \mathbb{Z}\\ \frac {a_1^2-a_1}{d}+2a_1p+p^2d\in \mathbb{Z}\]It is not difficult to show that \(a_1,d\in \mathbb{Q}\), thus we can assume \(\displaystyle a_1=\frac {x_1}{y_1}, d=\frac {x_2}{y_2}\) with \(x_1,y_1,x_2,y_2\in \mathbb{Z}\) and \((x_1,y_1)=1, (x_2,y_2)=1\). It suffices to show that \(y_1=y_2=\pm 1\).

Plugging these into the first expression gives \(\frac {(x_1^2-x_1y_1)y_2}{y_1^2x_2}\in \mathbb{Z}\iff y_1^2x_2|(x_1^2-x_1y_1)y_2\). We can simplify this further by considering the relatively prime condition. Note that \((y_1,x_1)=1\implies (y_1,x_1-y_1)=1\), thus \((y_1^2, x_1^2-x_1y_1)=1\). Hence \(\displaystyle y_1^2|y_2 (\star)\).

Applying the same substitution for the second equation gives \(y_1y_2|2px_1y_2+p^2x_2y_1\implies y_2|p^2x_2y_1\implies\) \(\displaystyle y_2|p^2y_1\). From \((\star\) we can let \(y_2=y_1^2k\) for some integer \(k\), thus \(y_2|p^2y_1\implies y_1^2k|p^2y_1\implies y_1k|p^2\). Similarly, the analogous operation on the third equation gives \(y_1k|q^2\). This means \(y_1k\) is a common divisor of \(p^2,q^2\). Since \((p^2,q^2)=1\), \(|y_1k|=1\implies |y_1|=1, |y_2|=1\), which is what we want to prove. – Xuming Liang · 8 months, 1 week ago

Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

I could reach to the equation (*) but then beated around the Bush after that how much will i get on this question then?Log in to reply

– Priyanshu Mishra · 8 months, 1 week ago

You can get 3/4 of the total marks of the question for your solution. If a question is of 16.6 marks then i think you should get around 12 marks.Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

Thanx for that....I just hope u r right.Log in to reply

– Priyanshu Mishra · 8 months, 1 week ago

In which class are you?Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

11thLog in to reply

– Priyanshu Mishra · 8 months, 1 week ago

Which books you studied for INMO?Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

Problem primer and rajeev manochaLog in to reply

Which books did you for both RMO and INMO?– Saurabh Mallik · 8 months, 1 week agoLog in to reply

– Samarth Agarwal · 8 months, 1 week ago

2 and a half or around that....I used problem primer and rajeev manochaLog in to reply

– Rohit Kumar · 8 months, 1 week ago

I could only prove a and d are rational. How much should I expect ?( I did mention that those 3 expressions were integers.)Log in to reply

– Shourya Pandey · 8 months, 1 week ago

I did the same. I really don't think there is any other solution to it. Nice write up, by the way.Log in to reply

– Priyanshu Mishra · 8 months, 1 week ago

Nice solution. Sequence problems are very fond for the problem posers as it requires many topics.Log in to reply

5TH QUESTION

Log in to reply

5TH QUESTION

Log in to reply

I am Anay Karnik from Mumbai, just received my scorecard, got 51. Let's make a list to determine the cut-off.

Anay Karnik (Mumbai) - 51

By the way, the question paper uploaded is actually mine (I know from the wrinkles on the bottom of the paper) I was the first one to reach home, and scan my paper and upload it on AOPS. The person who uploaded it here has copied mine from there! – Anay Karnik · 7 months, 3 weeks ago

Log in to reply

– Samarth Agarwal · 7 months, 3 weeks ago

How many questions have u attempted?Log in to reply

– Anay Karnik · 7 months, 3 weeks ago

4, no chance for me, cutoff will be 65 or above.Log in to reply

– Raushan Sharma · 7 months, 3 weeks ago

Oh, How can you say that?? 51 is quite a good scoreLog in to reply

– Anay Karnik · 7 months, 3 weeks ago

I just checked in with the list in AOPS where lot of people have posted their marks and some people are predicting 65 cut-off.Log in to reply

@Anay Karnik: 65 is too high.... it would be around 55.. – Abhishek Bakshi · 7 months, 2 weeks ago

Log in to reply

– Shrihari B · 7 months, 2 weeks ago

No 55 is too low for a cutoff this year. I agree with Anay Karnik. The cutoff will be 63-65. And hi Anay this is me Shrihari :)Log in to reply

– Raushan Sharma · 7 months, 2 weeks ago

No, I also saw that list in AOPS, yes Anay is right, cutoff might be around 65 this timeLog in to reply

@Shrihari B, @Raushan Sharma: The cutoff really doesn't matter much to me... but even if the paper is easy, the probability that the cutoff will go above 60 is really low... well anyway I hope you guys get selected whatever be the cutoff... :-D – Abhishek Bakshi · 7 months, 2 weeks ago

Log in to reply

– Svatejas Shivakumar · 7 months, 3 weeks ago

Yes I copied it from Aops. Don't remember who had posted it but I guess it must be you :P I had earlier mentioned in the note that I copied it from Aops as I didn't write INMO. Probably a moderator edited it. Anyway thanks for uploading the paper, had great fun solving or at least trying to solve the problems :)Log in to reply

– Anay Karnik · 7 months, 3 weeks ago

no problemLog in to reply

3.

(i)By the well ordering principle the set of positive integral values of \(T^k(n)\) must have a least value(positive integral) say m. Let's assume that \(m \neq 1\).

If m is even:\[T(m)=\dfrac{m}{2}<m\]

If m is odd:\[T(T(m))=\dfrac{m+1}{2}<m\]

Both cases lead to a contradiction. Hence our assumption is wrong. – Svatejas Shivakumar · 8 months, 1 week ago

Log in to reply

– Akash Deep · 8 months, 1 week ago

i too did exactly the same thingLog in to reply

A short proof to problem 4 is to consider 2017 very close points, say, the angle made by the minor arc formed by the farthest two points is less than \(\frac {2\pi}{n} \) . Complete the 2017 regular n-gons formed by these points. Clearly(one shouod explain this in the examination) no two n-gons have any points in common. Therefore, by the pigeonhole principle, at least one of these n-gons has no red point as its vertex. – Shourya Pandey · 8 months, 1 week ago

Log in to reply

– Abhishek Bakshi · 8 months ago

Hey, Shourya, tell me what's wrong in this proof. Let's label 2 adjacent red points to be \(A_1\) and \(A_2\). Now we divide the arc between the 2 point into an arbitrarily large number of equal parts, say, \(10^9\), and we mark points labelled as \(R_1\, R_2\) and so on to denote the boundary of the sectors. Now we draw a regular n-gon with \(A_1\) as one of the vertex. It may or may not contain any other red point. Now we rotate the n-gon so that the vertex shifts to \(R_1\). This n-gon also may have a red point on it. If it doesn't then this is our required polygon. If it does then we rotate it again to \(R_2\) and so on. We have only 2016 red points hence a maximum number of 2015 of the regular n-gons out of the \(10^9\) polygons so created will have red points as one of the vertices. Hence the others are our required polygons. Hence proved.Log in to reply

– Rohit Kumar · 7 months, 3 weeks ago

That's pretty much same to what Shourya did. I think it's correct.Log in to reply

– Aditya Kumar · 7 months, 3 weeks ago

I am not sure about it. But I feel that you should have proved that the the n-gons are regular. See the arrangement of the points is not regular in the case you have mentioned. So getting a regular n-gon will be a very special case. You need to prove that the special case exists.Log in to reply

– Abhishek Bakshi · 7 months, 3 weeks ago

Huh????? what are you talking about... you want an n-gon with blue vertices which are infinite in number in contrast to the red ones which are only 2016 in number. A regular n-gon with blue vertices is not a special case..Log in to reply

regular n-gon. For that you have to prove that the sides are equal and all angles are equal. Now according to your explanation, the \(R_{n}\) distribution is irregular. Hence obtaining a regular n-gon becomes a special case. – Aditya Kumar · 7 months, 3 weeks agoLog in to reply

– Abhishek Bakshi · 7 months, 3 weeks ago

I think you have misunderstood my solution..I am not saying that a regular n-gon will be formed using the points of \(R_n\). I am saying that just take one point of \(R_n\)and draw a regular n-gon with it as one of the vertices. Such a polygon always exists and we have to just prove that out of the \(10^9\) polygons such formed, the 2016 red points don't lie on all of them which is quite obvious. I believe that you understand the sarcasm too...haha..Log in to reply

this. – Aditya Kumar · 7 months, 3 weeks ago

Ooh sorry I actually misunderstood the solution. Meanwhile have a look atLog in to reply

@Aditya Kumar: what do u think about the above solution?? – Abhishek Bakshi · 7 months, 4 weeks ago

Log in to reply

– Rohit Kumar · 8 months, 1 week ago

Brilliant ! 17/17.Log in to reply

– Shourya Pandey · 8 months, 1 week ago

You too have a good solution to the problem.Log in to reply

A lengthy proof to problem 4: Let the circle given in the question be the unit circle centred at the origin, WLOG. Each point has a unique polar coordinate \( (1, \theta) \), where \( 0 \le \theta <2\pi \). We will express all points (on this circle) only by its polar angle. Let \(\theta_{i} , i= 1,2,3,...,2016 \) denote the 2016 red points on the circle. For convinience, denote \( \frac {\theta}{\pi} \) as \(b_i\)

LEMMA 1 :- There exists a number \(\alpha \), such that \( \alpha - b_i\) is irrational, for all \(i\). (Try proving this. I'll post the proof later).

Now, we choose the points of the regular n-gon to be \( \alpha\pi , \alpha\pi + \frac {2\pi}{n} , ..., \alpha\pi + \frac {2(n-1)\pi}{n} \). Here \(\alpha \) may not lie in \([0, 2\pi) \), but it does not matter.

Suppose some vertex of this n-gon coincides with some red point, say \( \alpha\pi + \frac {2l\pi}{n} = \theta_i+ 2k\pi\), where \(l\) is one of\(0,1,2,...,n-1\) and \(k\) is an integer. Then rearranging it gives \(\alpha - b_i= 2k - \frac {2l}{n} \), a contradiction to the lemma.

This solves the problem. – Shourya Pandey · 8 months, 1 week ago

Log in to reply

Anyone who cleared INMO? – Mayank Jha · 7 months ago

Log in to reply

– Shubham Jain · 7 months ago

I did (got 73). You should join the discussion on AoPS, my username is TheOneYouWant there... looks like someone below was impersonating me :PLog in to reply

– Shubham Jain · 6 months, 3 weeks ago

Thanks guys! A lot of people have been asking me the same question lately, thus I have written an article on it on my site w/ my friends: http://mathometer.weebly.com/preparing-for-inmo.html The site is a mini success, garnering 12k views in the 3 weeks of its existence :)Log in to reply

Congrats! Shubham on getting selected for INMO. Which standard you are currently in?? Can you tell/guide me on how did you prepare for INMO please??– Saurabh Mallik · 7 months agoLog in to reply

– Mayank Jha · 7 months ago

Pls can you advise me on how to prepare effectively for INMO. I am not in AOPS.Log in to reply

Official solutions are out! Click here – Svatejas Shivakumar · 7 months, 3 weeks ago

Log in to reply

I used simple co-ordinate geometry to solve question 1. Will that be considered? – Saswata Banerjee · 7 months, 3 weeks ago

Log in to reply

– Shourya Pandey · 7 months, 3 weeks ago

Yes. If your solution is entirely correct, full marks will be awarded.Log in to reply

– Rohit Kumar · 7 months, 3 weeks ago

Do they award full marks ?(I heard from someone that they don't award you full marks even if your solution is correct )Log in to reply

This is why it is often recommended to use coordinate only as a tool to prove small steps in the problems. Of course, it is different if you're very good at bashing. – Shourya Pandey · 7 months, 3 weeks ago

Log in to reply

– Rohit Kumar · 7 months, 3 weeks ago

In the sixth question, I was able to prove that a and d were rational. How much should I expect ?Log in to reply

Hey guys I would Like if everyone posted their expected marks here. That would help us to get a rough idea of the cutoff. Starting with me ... I am expecting 34 :( After u post your marks I will edit it in here and then delete your comments to avoid a lot of junk. – Shrihari B · 8 months, 1 week ago

Log in to reply

– Racchit Jain · 8 months, 1 week ago

A friend of mine got five and a half questions correct.Log in to reply

– Raushan Sharma · 8 months, 1 week ago

I am also expecting about 35 to 40.Log in to reply

2.

(I)Let \(a=b=1\). Therefore \(c^3-2c+1=0\). Solving we get a solution as \(c=\dfrac{-1+\sqrt{5}}{2}\). Hence it is not necessary for \(a=b=c\)

(II)We may assume \(a=max(a,b,c)\) since \(a(a^3+b^3)=b(b^3+c^3)\) and \(a \ge b\), \(a^3+b^3 \le b^3+c^3\) or \(c \ge a\). This forces \(a=c\). Similarly, \(a(a^3+b^3)=c(c^3+a^3)\) or \(b=c\).

Hence it is necessary for \(a=b=c\). – Svatejas Shivakumar · 8 months, 1 week ago

Log in to reply

– Shourya Pandey · 8 months, 1 week ago

The second part is NOT SYMMETRIC, but only CYCLIC. Therefore \(a \ge b \ge c\) and \(a \ge c \ge b\) are two different cases.Log in to reply

– Svatejas Shivakumar · 8 months, 1 week ago

Yes you are right. Is my solution correct now?Log in to reply

– Shourya Pandey · 8 months, 1 week ago

Indeed.Log in to reply

– Shrihari B · 8 months, 1 week ago

Hey Shourya I didn't take the second case i.e. a>=c>=b. How many marks will be deducted ???Log in to reply

– Aman Anand · 8 months, 1 week ago

I assumed symmetric and the first inequality you wrote .how much should I get?Log in to reply

– Svatejas Shivakumar · 8 months, 1 week ago

You cannot assume the symmetric case as Shourya mentioned. I am not sure how much you would get but I guess 6-7. Hard luck.Log in to reply

– Aman Anand · 8 months, 1 week ago

I solved 1,2,3 please see no. 3 which i posted above.I assumed a>=b>=c but not a>=c>=b in the second one and justified the first case of no. 2.how much should I get in no. 2.Log in to reply

– Svatejas Shivakumar · 8 months, 1 week ago

Am I correct?Log in to reply

– Raushan Sharma · 8 months, 1 week ago

Yes, absolutely correct. Just I assumed a=b=2, and then c=\(\sqrt{5} - 1\) for the 1st one, and 2nd one same assumption WLOG, and then some manipulation.Log in to reply

You know anyone from your region and how much he/she did? – Raushan Sharma · 8 months, 1 week ago

Log in to reply

– Svatejas Shivakumar · 8 months, 1 week ago

Sorry I don'tLog in to reply

– Raushan Sharma · 8 months, 1 week ago

OoohkLog in to reply

what are your marks – Deekshith Kanagala · 2 months, 3 weeks ago

Log in to reply

what is the cuttoff inmo 2016 – Deekshith Kanagala · 2 months, 3 weeks ago

Log in to reply

Got the scorecard....could only make 35.... :( – Samarth Agarwal · 7 months, 2 weeks ago

Log in to reply

I wanna ask is this correct for no. 3 T(2)=1 T(3)=4,=>T^(3)=1 T(4)=2,=>T^2(4)=1 1)We prove by induction.Assume for some 4 \leq m \geq 2n and some k,we can get T^k(m)=1. Case 1) for m=2n+1 T(m)=2n+2;=>T^2(m)=n+1 < 2n So we can get for some k, T^k(2n+1)=1 Similarly for case 2) .So this proves our assumption. 2) T^{k+2} (n)=1;for c

{k+2} values. Case 1)n=2m T^{k+2} (2m)=T^{k+1} (m)=1;for c{k+1} values. Case 2)n=2m-1 T^{k+2} (2m-1)=T^{k+1} (2m)=T^k (m)=1;for c{k} values. This shows c{k+2}=c{k+1}+c{k}. – Aman Anand · 8 months, 1 week agoLog in to reply

4.

Isn't this one very simple? There are infinitely many points which are blue. There are also infinitely many points which are blue between two red points. There are only 2016 points, so definitely there are infinitely many regular n-gons which means that there are infinitely many regular n-gons with blue vertices.

Am I missing something obvious? – Svatejas Shivakumar · 8 months, 1 week ago

Log in to reply

– Raushan Sharma · 8 months, 1 week ago

Yes, this question I couldn't understand what it meant, or what I needed to show, coz there are infinitely many points on the circle, so... what does this mean!!Log in to reply

– Shourya Pandey · 8 months, 1 week ago

Sort of. Just put it on paper vividly.Log in to reply

– Akash Deep · 8 months, 1 week ago

what do u think is this years paper easier or last yearsLog in to reply

– Shourya Pandey · 8 months, 1 week ago

I think the papers were comparable. Although the paper this year was slightly difficult on an average, but the 5th question this year wasn't impossible in the time-limit ( unlike last year's).Log in to reply

– Rohit Kumar · 8 months, 1 week ago

Are you sure ? Because 1st, 2nd and 4th were easy.(I might be wrong) How many did you solve ?Log in to reply

– Shourya Pandey · 8 months, 1 week ago

So were questions 4 and 6 last year.Log in to reply

– Shrihari B · 8 months, 1 week ago

Hi Shourya... what is the cutoff you predict for this year .... We need some expert advice :)Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

Pls tell the sol. Of 4thLog in to reply

If a vertex is already on a red point, measure the angle from adjacent red points.

Hence proved. – Rohit Kumar · 8 months, 1 week ago

Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

I m weak at combinotorics but this seems correct. ...btw how many did u got correct n in which class r u?Log in to reply

– Rohit Kumar · 8 months, 1 week ago

I solved 1st , 2nd, 4th completely, solved part 1 of 3rd, and I couldn't complete 6th. I am in 11th.Log in to reply

Can anybody please explain me question no. \(3\) properly?Rohit you belong to which region? Which books did you study for RMO and INMO??– Saurabh Mallik · 8 months, 1 week agoLog in to reply

– Raushan Sharma · 8 months, 1 week ago

Couldn't get it!! Can you please explain a bit thorough??Log in to reply

– Rohit Kumar · 8 months, 1 week ago

That's exactly the solution I wrote in the exam! Don't tell it's incomprehensible!Log in to reply

– Raushan Sharma · 8 months, 1 week ago

No, it isn't u will get 17 definitely. I understood now, but for that I had to draw figures and all, BTW, nice solutionLog in to reply

– Mayank Jha · 8 months, 1 week ago

Would you guide me to prepare for Rmo and INMO?Can you give me your mail address?Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

Are 2 and a half or around question enough for selectionLog in to reply

– Raushan Sharma · 8 months, 1 week ago

I have also solved 2 and a half, but I think that's not enough for selection, but yeah, we can expect being in the merit listLog in to reply

– Akash Deep · 8 months, 1 week ago

i did 1, 2 (in 2nd i just showed for the first case that all 3 of them cannot be distinct , means any 2 are equal , after taking this i showed that if a=b=x and x and c satisfy x^3 +c^3 - 2c^2x = 0 then there is no need for x = c)whereas the 2nd part of 2nd question i did correctly. in 3rd i did first part correctly and if s(k) denotes the set containing elements for which t^k(n) in ,2nd part i showed that s(k) is a subset of s(k+2) and showed that from every element of s(k+1) we can generate a corresponding element of s(k+2), then i showed that if there is any element in s(k+2) it is either of s(k) or generated from s(k+1) 4- did not attempt 5- wrote that R*S stuff and left it. 6. - just wrote that for an a.p to have all elements integal , c.d and first term must be integral with a useless proof of this. (please let me know if i have any chances and how much marks can i get)Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

Many people are getting only 2 to 3 questions correct so may be cutoff ho downLog in to reply

Does anyone knows what will be the cutoff this year – Samarth Agarwal · 8 months, 1 week ago

Log in to reply

– Aman Anand · 8 months ago

It will be three and half questions surely.Log in to reply

– Rohit Kumar · 8 months ago

I feel the same.Log in to reply

– Harsh Shrivastava · 8 months, 1 week ago

How many did you solved?Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

Only 2... :( btw how many did u solved?Log in to reply

btw which two? – Harsh Shrivastava · 8 months, 1 week ago

Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

1,3...and u?..I dont expect my selection now also we will not be eligible next year :'(Log in to reply

– Harsh Shrivastava · 8 months, 1 week ago

I can give it next year if i qualify RMO.(since i am in class 10)Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

I thought u are in class 11....but if u have done 2 then u may get inmo merit certificate and direct eligibility for inmo next yearLog in to reply

– Harsh Shrivastava · 8 months, 1 week ago

I don't think so, many people got 3 or more correct.Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

Do u know the solution top fifth one? ......using rs=area I reached somewhere and I needed only to prove that bd^2=area of abc.....not getting after thatLog in to reply

I tried to use coordinate bash in 5th but calculations were tooo lengthy so left it. – Harsh Shrivastava · 8 months, 1 week ago

Log in to reply

– Deekshith Kanagala · 8 months, 1 week ago

yes it is too long ,but you can prove using stewarts theorm rightLog in to reply

– Samarth Agarwal · 8 months, 1 week ago

Pls can u elaborateLog in to reply

r'(AB+BD+AD)/2+r'(BD+CD+BC)/2=r(AB+BC+AC)/2

On solving (1/r')=(1/r)+(BD/area of ABC)

I was struck after that now we only need to prove BD^2=ac/2 but how?..... – Samarth Agarwal · 8 months, 1 week ago

Log in to reply

– Mayank Jha · 8 months, 1 week ago

Use similarity of triangles which may helpLog in to reply

– Harsh Shrivastava · 8 months, 1 week ago

Clever way!Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

But I could not reach the endLog in to reply

– Svatejas Shivakumar · 8 months, 1 week ago

How did you solve question 3? I think that I am getting only part 1 in it.Log in to reply

– Raushan Sharma · 8 months, 1 week ago

I also got the first part fully, but in the second part I tried many things, but didn't land up with the result. What do u think is the partition of marks in Q.3??Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

Since i am using brilliant on phone its difficult to use latex once my pc is repaired i would write a sol. To 3rd one ....its very interestingLog in to reply

– Raushan Sharma · 8 months, 1 week ago

Ok, I would surely like to see thatLog in to reply

– Samarth Agarwal · 8 months, 1 week ago

I would soon write a complete sol. Of ques. 3.... the second part involves cases taking a no. In the set to be once even and then odd... (I would post complete solution soon)Log in to reply

– Deekshith Kanagala · 8 months, 1 week ago

if the length of the cevian is <p> then the ratio it divides the hypotenuse is a+p/c+p.where c,a are its sidesLog in to reply

– Raushan Sharma · 8 months, 1 week ago

Yes, I also got to the same to prove BD^2 = Area, then I was trying with Stewart's theorem, and also dropped perpendiculars from A and C on BD, and tried to find the value of BD^2 in two different ways, and equate, but the time was up, and it was incomplete LOL. I think this problem was quite manipulative.Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

So how much we sud expect in this question. ...btw in which class u rLog in to reply

– Raushan Sharma · 8 months, 1 week ago

I think we can get about 10 marks in this question. I am in 10th.Log in to reply

– Samarth Agarwal · 8 months, 1 week ago

10 is very much for 3 statements....may be 5?......but i just hope u r rightLog in to reply

– Raushan Sharma · 8 months, 1 week ago

No, no, actually one of my fellow mates Vishal Raj solved it fully using Stewart's theorem only, and his strategy was also the same as writing BD^2 in two different ways, and then manipulating. So, we can expect 10 marks I hopeLog in to reply

– Samarth Agarwal · 8 months, 1 week ago

Hope so, fingers crossedLog in to reply

Log in to reply

– Svatejas Shivakumar · 8 months, 1 week ago

Try zooming in.Log in to reply

– Svatejas Shivakumar · 8 months, 1 week ago

I am able to see it clearly. If you are still not able to see the problems, I will type the problems here.Log in to reply

– Aditya Kumar · 8 months, 1 week ago

Yes now it is visible. Thanks.Log in to reply

– Deekshith Kanagala · 7 months, 3 weeks ago

did anybody got the score cardLog in to reply

@deekshith kanagala: how much are you expecting... I think this year's cutoff will be higher than that of last year... it should be around 55.. those who get selected would have solved the 1st, 2nd and 4th problems fully and 1st part of the 3rd problem... – Abhishek Bakshi · 7 months, 3 weeks ago

Log in to reply

– Deekshith Kanagala · 7 months, 3 weeks ago

i solve 1st and 3rd fully.i think i will get full marks for these two questonsLog in to reply

– Samarth Agarwal · 7 months, 3 weeks ago

I also did 1st and 3rd full and did 5th,6th partially......is that enough for qualifying?...has anybody received the scorecards?Log in to reply

– Deekshith Kanagala · 7 months, 3 weeks ago

what is the cutoff ?.is it more than the previous year?Log in to reply

– Samarth Agarwal · 7 months, 3 weeks ago

Cant predict...let the scorecards comeLog in to reply

– Muralidaran C · 7 months, 3 weeks ago

Looks like the performance cards have been dispatched for the Mumbai region!Log in to reply

– Deekshith Kanagala · 7 months, 3 weeks ago

you are in 11th standard right?Log in to reply

– Samarth Agarwal · 7 months, 3 weeks ago

Yes...in which standard are you?Log in to reply

– Deekshith Kanagala · 7 months, 3 weeks ago

i am also in 11th.did you wrote inmo 2015?Log in to reply

– Samarth Agarwal · 7 months, 3 weeks ago

No, I could not qualify RMO...in which coaching do you go?Log in to reply

– Deekshith Kanagala · 7 months, 3 weeks ago

i got the score card. i got 35Log in to reply

@deekshith kanagala: nice score... you are quite accurate as you got the marks for 2 questions.... but i don't think the cutoff will go so low... btw I am in 12th standard and i gave inmo last 2 years... unfortunately i missed the cutoff last year by 6 marks, so i have a little knowledge about what the cutoff could be... – Abhishek Bakshi · 7 months, 2 weeks ago

Log in to reply

I am getting 65 – The one You want · 7 months, 3 weeks ago

Log in to reply

– Shourya Pandey · 7 months, 3 weeks ago

Hi. It is great to see this. Which region are you from, and what is your name?Log in to reply

– Raushan Sharma · 7 months, 3 weeks ago

You got your scorecard??Log in to reply