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RMO 2016 practice board

Hello everyone!

As many of us are preparing for RMO (Regional Mathematics Olympiad), let us start posting problems and help each other prepare. Everyone is more than welcome to post problems or post the solutions to problems.

Here is a problem to start with:

In \(\Delta ABC\), \(O\) is the circumcenter and \(H\) is the orthocenter. If \(AO=AH\), prove that \(\angle A=60^\circ\).

Also, if the circumcircle of \(\Delta BOC\) passes through H, prove that \(\angle A=60^\circ\).

Note by Svatejas Shivakumar
3 months, 3 weeks ago

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Solution

1)Extend BO to meet at the circumcircle of the ∆ at M.Also extend CH to meet at AB at N and AH meet BC at K.Join AM and MC.

Now observe that angle ANC = angle BAM = 90°.This implies AM || CN.

Similarly,AH || CM.

This implies,AHMC is a parallelogram.

Now, AH = MC(=OA because AH=AO.)

Thus,OMC is an equilateral triangle with angle MOC =60°.

angle BOC = 180° - angle MOC = 120°.

This implies angle BAC = 60° Harsh Shrivastava · 3 months, 2 weeks ago

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@Harsh Shrivastava can u post the solution for the 2nd part of problem 1 Neel Khare · 3 months, 1 week ago

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@Neel Khare I will try to solve it and if I succeed ,I'll post the solution. Harsh Shrivastava · 3 months, 1 week ago

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@Harsh Shrivastava i got it here is the solution [url=https://postimg.org/image/e0cnnt8w5/][img]https://s18.postimg.org/e0cnnt8w5/IMG_0100.jpg[/img][/url] Neel Khare · 3 months, 1 week ago

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@Neel Khare sorry https://postimg.org/image/ylrffpqh1/ https://postimg.org/image/e0cnnt8w5/ open the links Neel Khare · 3 months, 1 week ago

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@Harsh Shrivastava the 1 st part can be solved much much easily AO=AH R=2RcosA 2cosA=1 cosA=1/2 A=60 done!! Neel Khare · 3 months, 1 week ago

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@Neel Khare Oh that's awesome use of trignometery! Harsh Shrivastava · 3 months, 1 week ago

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@Harsh Shrivastava I can't edit my comment, its "AHCM". Harsh Shrivastava · 3 months, 2 weeks ago

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@Harsh Shrivastava Brilliant staff are working on this issue. Sharky Kesa · 3 months, 2 weeks ago

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@Sharky Kesa Hey Sharky post some problems (RMO level.) Harsh Shrivastava · 3 months, 2 weeks ago

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In a triangle \(ABC\) the point \(D\) is the intersection of the interior angle bisector of \(\angle BAC\) with side \(BC\). The line through \(A\) that is perpendicular to \(AD\) intersects the circumcircle of triangle \(ABC\) for a second time at point \(P\). A circle through points \(A\) and \(P\) intersects line segment \(BP\) internally in \(E\) and line segment \(CP\) internally in \(F\).

Prove \(\angle DEP = \angle PFD\). Sharky Kesa · 3 months, 2 weeks ago

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@Sharky Kesa I have seen this question earlier in one of my books.

Its a good problem. As i have solution of it, i will not post this now. Let others try also. Priyanshu Mishra · 3 months, 2 weeks ago

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In \(\Delta ABC\), O is the circumcenter and H is the orthocenter. Prove that \(AH^2+BC^2=4AO^2\). Svatejas Shivakumar · 3 months, 2 weeks ago

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@Svatejas Shivakumar Can you post the solution please ? Alan Joel · 3 months, 1 week ago

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@Alan Joel i got it , its easy https://postimg.org/image/5d1sm6hm7/ Neel Khare · 3 months, 1 week ago

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@Neel Khare How did you get AH^2 = 2RcosA ? Alan Joel · 3 months, 1 week ago

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@Alan Joel Its an identity. Priyanshu Mishra · 3 months, 1 week ago

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@Priyanshu Mishra Its AH* Alan Joel · 3 months, 1 week ago

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@Alan Joel Oh, I knew that but I forgot lol Alan Joel · 3 months, 1 week ago

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Well, this first part of the question can't be right! (Below is a summary of why)

If \(AO=OH\), \(H\) must also be on the circumcircle of \(ABC\), from which we get the triangle being right-angled, and \(H\) is on the vertex with right angle. Nothing else can be gathered from the given information.

Perhaps you meant \(AO=AH\), which makes sense. Sharky Kesa · 3 months, 3 weeks ago

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@Sharky Kesa Yes you are right. Thanks for pointing it out. Svatejas Shivakumar · 3 months, 2 weeks ago

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@Svatejas Shivakumar, @Harsh Shrivastava, @Ayush Pattnayak @Alan Joel @Racchit Jain @rajdeep das @naitik sanghavi and all other RMO aspirants ,I invite you'll to my RMO,INMO team. Those who are interested can give their email id over here. Ayush Rai · 2 days, 19 hours ago

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@Ayush Rai Mine is rajdeep.ind24@gmail.com Rajdeep Das · 2 days, 13 hours ago

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@Rajdeep Das Ok i have sent you an invite You can check your email

By the way I am the co owner of the team and ayush is the owner Neel Khare · 2 days, 12 hours ago

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@Ayush Rai I'm in, here's my email id alanj.33@cloud.com Alan Joel · 2 days, 19 hours ago

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@Alan Joel I meant icloud* there Alan Joel · 2 days, 19 hours ago

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@Alan Joel you can check ur mail now Ayush Rai · 2 days, 19 hours ago

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Hello everybody,

RMO results are out.

Who are selected? Priyanshu Mishra · 1 month, 1 week ago

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Use trigonometry as there is a right triangle formed assuming centet Biswajit Barik · 1 month, 1 week ago

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Hey!! Did anyone give GMO? Or RMO on 16th October. If yes please tell how many were you able to do, and what should be the expected cutoff Racchit Jain · 3 months ago

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@Racchit Jain What is your score in RMO? Sayantan Saha · 2 months, 3 weeks ago

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@Sayantan Saha I gave GMO Racchit Jain · 2 months, 3 weeks ago

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@Racchit Jain However, I know marks of some of my friends from different regions, which region are you asking for? Racchit Jain · 2 months, 3 weeks ago

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@Racchit Jain Pls give for delhi region also. Rajdeep Das · 2 months, 3 weeks ago

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@Rajdeep Das Is RMO DELHI result out? Priyanshu Mishra · 2 months, 3 weeks ago

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@Priyanshu Mishra Yes Rajdeep Das · 2 months, 3 weeks ago

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@Rajdeep Das At which website? Priyanshu Mishra · 2 months, 3 weeks ago

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@Racchit Jain I am from WB region. I want to know how high the scores of rmo had gone in Delhi this year. Sayantan Saha · 2 months, 3 weeks ago

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@Sayantan Saha The highest marks in Delhi that I know of is 35 out of 60 otherwise everyone is getting less than 15. The cutoff should be around 20 I think, but not more than 25 Racchit Jain · 2 months, 2 weeks ago

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@Racchit Jain Only 35. I don't think so.

At which website is the result of RMO declared? Priyanshu Mishra · 2 months, 2 weeks ago

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Given are two circles w1, w2 which intersect at points X, Y . Let P be an arbitrary point on w1. Suppose that the lines PX, PY meet w2 again at points A,B respectively. Prove that the circumcircles of all triangles PAB have the same radius. Sayantan Saha · 3 months, 1 week ago

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@Sayantan Saha this is north zone's (Delhi) problem Sayantan Saha · 3 months, 1 week ago

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@Sayantan Saha could u do it? Rajdeep Das · 3 months, 1 week ago

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@Rajdeep Das Then pls post the solution Rajdeep Das · 3 months, 1 week ago

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@Rajdeep Das Show that AB is independent of the choice of point P Racchit Jain · 3 months, 1 week ago

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@Rajdeep Das Try using sine rule Racchit Jain · 3 months, 1 week ago

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@Racchit Jain It looks like Power of a Point, but Extended Sine Rule definitely works. Sharky Kesa · 3 months, 1 week ago

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@Sharky Kesa please add the solution Sayantan Saha · 3 months, 1 week ago

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@Sayantan Saha I won't give solution but the crux of this proof is to show that \(AB\) is constant, irrespective of where \(P\) is. Sharky Kesa · 3 months, 1 week ago

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@Sharky Kesa yeah. I have done it Sayantan Saha · 3 months, 1 week ago

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@Sayantan Saha Two circles C1 and C2 intersect each other at points A and B. Their external common tangent (closer to B) touches C1 at P and C2 at Q. Let C be the reflection of B in line PQ. Prove that angleCAP = angleBAQ. Can you convince me what this reflection does mean. Sayantan Saha · 3 months, 1 week ago

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Please someone post this years rmo question paper. Harsh Shrivastava · 3 months, 1 week ago

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@Harsh Shrivastava The papers are uploaded on AoPS. Svatejas Shivakumar · 3 months, 1 week ago

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@Svatejas Shivakumar Please give me the link.Thanks. Harsh Shrivastava · 3 months, 1 week ago

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@Harsh Shrivastava I have posted Gujarat rmo paper here,https://brilliant.org/discussions/thread/rmo-2016-gujarat-region/?ref_id=1272714 Naitik Sanghavi · 3 months, 1 week ago

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Can someone post this year's problems? Sharky Kesa · 3 months, 1 week ago

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Mumbai region paper was really easy Racchit Jain · 3 months, 1 week ago

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@Racchit Jain Could you post the question paper please? Kush Singhal · 3 months, 1 week ago

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@Kush Singhal Yeah sure, but I use the app and I don't know how to post an image here, can you give me your email and I'll mail it to you? Racchit Jain · 3 months, 1 week ago

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@Racchit Jain Post it on Slack. Sharky Kesa · 3 months, 1 week ago

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@Sharky Kesa Umm...how do I do that? Racchit Jain · 3 months, 1 week ago

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@Racchit Jain It's asking me too get the app can't I do it using the browser only? Racchit Jain · 3 months, 1 week ago

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@Racchit Jain You can do it on the browser. Sharky Kesa · 3 months, 1 week ago

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@Sharky Kesa Can I mail it to you and then you can post it? Racchit Jain · 3 months, 1 week ago

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@Racchit Jain Sure. sharkesa@gmail.com Sharky Kesa · 3 months, 1 week ago

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@Sharky Kesa Please post the paper. Harsh Shrivastava · 3 months, 1 week ago

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@Sharky Kesa I have sent it to you plz check Racchit Jain · 3 months, 1 week ago

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Did anyone give RMO from north zone? Rajdeep Das · 3 months, 1 week ago

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@Rajdeep Das Uttar Pradesh - Me Rishik Jain · 3 months, 1 week ago

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@Rishik Jain At which center? Rajdeep Das · 3 months, 1 week ago

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@Rajdeep Das Meerut Rishik Jain · 3 months, 1 week ago

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Best of luck everyone Rajdeep Das · 3 months, 1 week ago

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Suppose that \(k, n_1, \ldots, n_k\) are variable positive integers satisfying \(k \geq 3\), \(n_1 \geq n_2 \geq \ldots \geq n_k \geq 1\), and \(n_1 + n_2 + \ldots + n_k = 2016\).

Find the maximal value of

\[\displaystyle \sum_{i=1}^{\left \lfloor \frac{k}{2} \right \rfloor + 1} \left ( \left \lfloor \dfrac {n_i}{2} \right \rfloor + 1 \right ) . \] Sharky Kesa · 3 months, 2 weeks ago

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@Sharky Kesa Please post again as you cannot edit that. Priyanshu Mishra · 3 months, 2 weeks ago

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@Priyanshu Mishra Sure I can! Mod powers! :P Sharky Kesa · 3 months, 2 weeks ago

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@Sharky Kesa But how you edited that? Priyanshu Mishra · 3 months, 2 weeks ago

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@Priyanshu Mishra With great skill (and a large screen)! Sharky Kesa · 3 months, 2 weeks ago

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@Sharky Kesa What is that skill!? Priyanshu Mishra · 3 months, 2 weeks ago

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@Priyanshu Mishra Big iMac skills! :P :P Sharky Kesa · 3 months, 2 weeks ago

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\(\large\ \frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}=x^2-11x-4\).

Find the largest real solution to this equation. Priyanshu Mishra · 3 months, 2 weeks ago

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Determine all positive triplets of integers such that

\(\large\ {(x+1)}^{y+1} + 1 = {(x+2)}^{z+1}.\) Priyanshu Mishra · 3 months, 2 weeks ago

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@Everyone

Find the smallest positive number \(\lambda\), such that for any complex numbers \({z_1},{z_2},{z_3} \in \{z\in \mathbb C \big| |z| < 1\}\), if \(z_1+z_2+z_3 = 0\), then \( \left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 < \lambda\).

Solve this and provide the solution. Priyanshu Mishra · 3 months, 2 weeks ago

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3) Let \(k\) be an integer and let

\[n=\sqrt[3]{k+\sqrt{k^2-1}} + \sqrt[3]{k-\sqrt{k^2-1}}+1\]

Prove \(n^3 - 3n^2\) is an integer. Sharky Kesa · 3 months, 2 weeks ago

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@Sharky Kesa Hint : Use the fact that if a+b+c = 0 then \(a^3+b^3+c^3 = 3abc\) Harsh Shrivastava · 3 months, 2 weeks ago

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@Harsh Shrivastava Good choice from MATHEMATICAL OLYMPIAD TREASURES. Priyanshu Mishra · 3 months, 2 weeks ago

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@Sharky Kesa \((n-1)^3=(\sqrt[3]{k+\sqrt{k^2-1}} + \sqrt[3]{k-\sqrt{k^2-1}})^3 \\ n^3-3n^2=2k-21\)

As \(k\) is integer \(2k-2\) will also be an integer. Akshat Sharda · 3 months, 2 weeks ago

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@Akshat Sharda Good but take \(n\) on RHS and see that \(a + b + c = 0\), so \(a^3 + b^3 + c^3 = 3abc\) and we are done. Priyanshu Mishra · 3 months, 2 weeks ago

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@Akshat Sharda Firstly, your final statement is incorrect. Secondly, you have put no working. Sorry, but this is a null solution. Sharky Kesa · 3 months, 2 weeks ago

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@Sharky Kesa Sorry but this is not a REGIONAL MATHEMATICAL OLYMPIAD level problem.

Please post difficult ones. Priyanshu Mishra · 3 months, 2 weeks ago

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@Priyanshu Mishra OK, sorry! I'll post IMO level probs next time. Sharky Kesa · 3 months, 2 weeks ago

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Prove that \[\dfrac{a^2+b^2+c^2}{d^2} >\dfrac{1}{3} \], where a,b,c,d are the sides of a quadrilateral. Harsh Shrivastava · 3 months, 2 weeks ago

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@Harsh Shrivastava This question I have done before (I'm pretty sure it was an application of QM-AM), so I'm leaving it as an exercise for everyone else. Sharky Kesa · 3 months, 2 weeks ago

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@Sharky Kesa I have a solution using trivial inequalities. Harsh Shrivastava · 3 months, 2 weeks ago

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@Harsh Shrivastava QM-AM is trivial. Sharky Kesa · 3 months, 2 weeks ago

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@Sharky Kesa Alright.Post some problems. Harsh Shrivastava · 3 months, 2 weeks ago

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@Sharky Kesa Yes after QM-AM the result directly follows. Svatejas Shivakumar · 3 months, 2 weeks ago

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I am also giving the RMO this year, please help me out 😀 Alan Joel · 3 months, 3 weeks ago

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What is circumdiameter? Harsh Shrivastava · 3 months, 3 weeks ago

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@Harsh Shrivastava sorry it is circumcenter. Svatejas Shivakumar · 3 months, 3 weeks ago

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@Harsh Shrivastava Sorry guys, I wasn't active on brilliant (and might not be for a period of time) Best luck for your rmos Nihar Mahajan · 3 months, 1 week ago

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@Nihar Mahajan Actually RMO happened today for most of the regions. Some are on 16th. Svatejas Shivakumar · 3 months, 1 week ago

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@Svatejas Shivakumar yep, i gave it today :) Nihar Mahajan · 3 months, 1 week ago

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@Nihar Mahajan Please most the paper. Harsh Shrivastava · 3 months, 1 week ago

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@Harsh Shrivastava How to most a paper? 😂😂 XD Nihar Mahajan · 3 months, 1 week ago

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@Nihar Mahajan Lol I meant post.Please post your paper. Harsh Shrivastava · 3 months, 1 week ago

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@Nihar Mahajan How was it? Svatejas Shivakumar · 3 months, 1 week ago

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@Svatejas Shivakumar How was ur rmo?how many did get correct? Naitik Sanghavi · 3 months, 1 week ago

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@Naitik Sanghavi Please most the paper. Harsh Shrivastava · 3 months, 1 week ago

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@Naitik Sanghavi Mine is on 16th. How was your paper? Svatejas Shivakumar · 3 months, 1 week ago

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@Svatejas Shivakumar 3-4 correct ,This time 2 question were very easy, so may be cutoff will go high! Naitik Sanghavi · 3 months, 1 week ago

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@Naitik Sanghavi From which region did u give RMO? Rajdeep Das · 3 months, 1 week ago

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@Rajdeep Das Gujarat Naitik Sanghavi · 3 months, 1 week ago

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@Svatejas Shivakumar pretty good, better than last time's Nihar Mahajan · 3 months, 1 week ago

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@Harsh Shrivastava Thanks. But I left Olympiad mathematics forever. Swapnil Das · 3 months, 3 weeks ago

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@Swapnil Das Are you ok? Say no Nihar Mahajan · 3 months, 1 week ago

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@Nihar Mahajan Yes, I'm fine.

Number Theory, Euclidean Geometry, Classical Inequalities...

Do they have any important application in my life? No, never. On the other hand, what I learn for the physics Olympiads will certainly have a huge impact on my future. Moreover, MOs make me slow, which is very harmful for these upcoming 3 years of my life. I'll be learning Math of relevant context like Calculus and stuff for PhOs, which will keep me away from MOs as well as keep my interest for math always alive. Swapnil Das · 3 months, 1 week ago

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@Swapnil Das WHATT!!!!!!!!!!!!! Harsh Shrivastava · 3 months, 3 weeks ago

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@Harsh Shrivastava That's true. Swapnil Das · 3 months, 3 weeks ago

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Here's the link to last year's RMO board Harsh Shrivastava · 3 months, 3 weeks ago

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RMO is over now. So no need to look at that. Now we should prepare for INMO.

I have posted a sample of 6 questions here. You can practice that and post more questions there also.

https://brilliant.org/discussions/thread/inmo-2017-board/. Priyanshu Mishra · 3 months, 1 week ago

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@Priyanshu Mishra It is not over in many regions, mine is on 16th. Harsh Shrivastava · 3 months, 1 week ago

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@Harsh Shrivastava Even mine is on 16th Alan Joel · 3 months, 1 week ago

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@Harsh Shrivastava How was your rmo :)? Nihar Mahajan · 3 months ago

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@Nihar Mahajan Well it could not have been any more bad.

Very very bad. :(

Can do only one question, I succumbed to exam pressure :(

Though I could solve 3 out of remaining 5 question at home myself.

I know this this is a lame excuse but 😭

My Olympiad maths is officially over.

Sorry for long reply, wbu? Harsh Shrivastava · 3 months ago

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@Harsh Shrivastava Your Olympiad maths journey is not over. Congo :) Nihar Mahajan · 1 month, 1 week ago

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@Nihar Mahajan Also,how's ya' iit prep going on?

How was KVPY? Harsh Shrivastava · 1 month, 1 week ago

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@Nihar Mahajan It might have been over if i was in a state like yours.

But i think we should do such math when we are free 'coz we enjoy olympiad maths.

Can you please suggest me some resources for inmo level geometry and some important topics in geometry to be studied? Harsh Shrivastava · 1 month, 1 week ago

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@Harsh Shrivastava Do you get selected in RMO?

I got selected. Priyanshu Mishra · 1 month, 1 week ago

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@Priyanshu Mishra Yes.Let's start the INMO Board!! Harsh Shrivastava · 1 month, 1 week ago

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@Harsh Shrivastava Oh congratulations.

I have already started that. Check here.

https://brilliant.org/discussions/thread/inmo-2017-board/?ref_id=1273098 Priyanshu Mishra · 1 month, 1 week ago

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@Priyanshu Mishra I think you should post a new note because that thread has died. Harsh Shrivastava · 1 month, 1 week ago

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@Harsh Shrivastava Ok i will post a fresh one. Priyanshu Mishra · 1 month, 1 week ago

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@Harsh Shrivastava Wth! Was the paper reallyl that tough? I solved 4 completely. I am on the boundary line of getting selected, since it's very difficult to get selected from my state, lot of competition here! 😅

Now it's ok, Harsh, I know you are gonna rock in other exams 😀😀 Nihar Mahajan · 3 months ago

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@Nihar Mahajan Did you give from Mumbai region? Racchit Jain · 2 months, 3 weeks ago

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@Nihar Mahajan Not sure about that rocking part :( Harsh Shrivastava · 3 months ago

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@Harsh Shrivastava So wait upto that and then solve INMO problems. Priyanshu Mishra · 3 months, 1 week ago

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@Priyanshu Mishra I have RMO on 23rd. Ayush Pattnayak · 3 months, 1 week ago

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