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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 Brilliant Member
1 year 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 - 1 year ago

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can u post the solution for the 2nd part of problem 1

Neel Khare - 1 year ago

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I will try to solve it and if I succeed ,I'll post the solution.

Harsh Shrivastava - 1 year 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 - 1 year ago

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@Neel Khare sorry https://postimg.org/image/ylrffpqh1/ https://postimg.org/image/e0cnnt8w5/ open the links

Neel Khare - 1 year 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 - 1 year ago

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@Neel Khare See an even more efficient use of \(trigonometry\). 1st Part of @Svatejas Shivakumar 's question :

\(1.\) Draw \(OD\perp BC\).

\(AO\) = \(AH\) => \(BO\) = \(2OD\) => \(\cos\) \(\angle BOD\) = \(1/2\) => \(\angle BOD\) = \(60\) => \(\angle BOC\) = \(120\) => \(\angle A\) = \(60\).

Second part :

\(2\) Quad. \(BHOC\) is cyclic.

=> \(\angle BHC\) = \(\angle BOC\)

=> \(180 - \angle A\) = \(2 \angle A\)

=> \(\angle A\) = \(180 / 3\) = \(60\).
I know, as expected it was a \(non-trigonometric\) one.

Rohit Camfar - 6 months, 2 weeks ago

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@Rohit Camfar please explain how BO = 2 OD if AO = AH. Thanks!

Yash Mehan - 3 months, 4 weeks ago

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@Yash Mehan That's just one of the Euler line properties. In a triangle \(\Delta ABC\), if \(O\) is the circumcenter, \(H\) is the orthocentre and \(D\) is the foot of perpendicular from \(O\) on \(BC\) then by a well known result \(AH = 2OD\).

Rohit Camfar - 3 months, 4 weeks ago

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@Neel Khare Oh that's awesome use of trignometery!

Harsh Shrivastava - 1 year ago

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I can't edit my comment, its "AHCM".

Harsh Shrivastava - 1 year ago

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Brilliant staff are working on this issue.

Sharky Kesa - 1 year ago

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@Sharky Kesa Hey Sharky post some problems (RMO level.)

Harsh Shrivastava - 1 year 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 - 1 year ago

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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 - 1 year ago

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Below is the \(diagram\).

\(Const:\) Let \(\odot APE\) be the circle passing through \(A\) and \(P.\)[where \(E\) is a point on \(BP\).] Join \(AE\), \(AF.\)

\(Solution\): Let \(\angle BAD\) = \(\angle CAD \) = \(\theta\).

Then \(\angle PAC\) = \(\angle PBC\) = \(\angle PCB\) = \(90\) - \(\theta\) & \(\angle BPC\) = \(\angle EPF\) = \(\angle EAF\) = \(2\theta.\)

Now, \(\angle BAC\) = \(\angle EAF\) => \(\angle BAE\) = \(\angle CAF\). Also, \(\angle ABE\) = \(\angle ACF\)

=> \(\Delta ABE\sim \Delta ACF\) => \(\dfrac{AB}{AC}\) = \(\dfrac{BE}{CF}\) ( Similarity properties )

Now, \(\angle EBD\) = \(\angle FCD\) = \(90\) - \(\theta\) , \(\dfrac{BE}{CF}\) = \(\dfrac{AB}{AC}\). But also \(\dfrac{BD}{CD}\) = \(\dfrac{AB}{AC}\) => \(\dfrac{BE}{CF}\) = \(\dfrac{BD}{CD}\).

=> \(\Delta EBD\sim \Delta FCD\).

=> \(\angle BED\) = \(\angle CFD\) =>\(\angle DEP = \angle PFD\).

\(KIPKIG.\)

Rohit Camfar - 6 months, 2 weeks ago

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How do you draw diagrams on the computer? Is there some tool you use?

Yash Mehan - 3 months, 4 weeks ago

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@Yash Mehan yes ..like geogebra

Ayush Rai - 3 months, 4 weeks ago

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Rohit Camfar - 6 months, 2 weeks ago

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In \(\Delta ABC\), O is the circumcenter and H is the orthocenter. Prove that \(AH^2+BC^2=4AO^2\).

Brilliant Member - 1 year ago

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Draw \(OD\perp BC\).

Just \(Pythagoras\) then, \(BD^{2}\) \(+\) \(OD^{2}\) = \(BO^{2}\)

=> \(4\) \(BD^{2}\) \(+\) \(4\) \(OD^{2}\) = \(4\) \(BO^{2}\)

=> \([2BD]^{2}\) \(+\) \([2OD]^{2}\) = \(4\) \(BO^{2}\)

=> \(BC^{2}\) + \(AH^{2}\) = \(4\) \(AO^{2}\).

Rohit Camfar - 6 months, 2 weeks ago

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Can you post the solution please ?

Alan Joel - 1 year ago

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i got it , its easy https://postimg.org/image/5d1sm6hm7/

Neel Khare - 1 year ago

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@Neel Khare How did you get AH^2 = 2RcosA ?

Alan Joel - 1 year ago

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@Alan Joel Its an identity.

Priyanshu Mishra - 1 year ago

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@Priyanshu Mishra Its AH*

Alan Joel - 1 year ago

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@Alan Joel Oh, I knew that but I forgot lol

Alan Joel - 1 year 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 - 1 year ago

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Yes you are right. Thanks for pointing it out.

Brilliant Member - 1 year ago

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@Swapnil Das @Harsh Shrivastava @Ayush Pattnayak Please Help! I am a class 9 student and am Appearing for RMO. I am pretty intimidated by Geometry Problems.... I can make an accurate figure but I don't know how to proceed.(For example: This question by Brilliant Member) Please guide me on how to solve Geometry and geometrical proofs...... I know Theorems(like Menelaus' Ptolemy's, Sine rule, Co-sine rule) If I make it in INMO... You all will deserve the credit. Urgent Help Required!!

Rishabh Gaharwar - 1 week, 4 days ago

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Just read those theorems and get into actual problem solving, you may also try British mathematical Olympiad problems, they are also RMO level. No sort of 'magical' construction required for RMO. Best of luck!

Harsh Shrivastava - 1 week, 4 days ago

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u just need some angle chasing and similarity for RMO.

Ayush Rai - 1 week, 4 days ago

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All INMO participants,please share ur marks.

Ayush Pattnayak - 8 months, 2 weeks ago

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70-80.

Priyanshu Mishra - 8 months, 2 weeks ago

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why dont u be active in slack?

Ayush Rai - 8 months, 2 weeks ago

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@Ayush Rai I don't have time for these "SLACK" things.

I have a lot of stuffs for FIITJEE. I do that only.

Priyanshu Mishra - 8 months, 2 weeks ago

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@Priyanshu Mishra Which centre?

Rajdeep Das - 8 months, 1 week ago

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@Priyanshu Mishra oh...ok I am very sorry for disturbing u.

Ayush Rai - 8 months, 2 weeks ago

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@Ayush Rai Its true. Kuch mazaa nahi aata slack chat pe.

Priyanshu Mishra - 8 months, 2 weeks ago

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@Priyanshu Mishra Its not chatting.I invited bcoz ur an INMO qualifier and u can help us solve problems that are posted.

Ayush Rai - 8 months, 2 weeks ago

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@Ayush Rai Initially i am only RMO qualifier. I am not INMO qualifier till the result is declared.

Priyanshu Mishra - 8 months, 2 weeks ago

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do u want to join my INMO team?

Ayush Rai - 8 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 - 9 months ago

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Me too.... gaharwar.02@gmail.com

Rishabh Gaharwar - 1 week, 4 days ago

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Me too.....ayushpattnayak2001@gmail.com

Ayush Pattnayak - 7 months, 2 weeks ago

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Mine is rajdeep.ind24@gmail.com

Rajdeep Das - 9 months ago

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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 - 9 months ago

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I'm in, here's my email id alanj.33@cloud.com

Alan Joel - 9 months ago

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I meant icloud* there

Alan Joel - 9 months ago

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@Alan Joel you can check ur mail now

Ayush Rai - 9 months ago

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Hello everybody,

RMO results are out.

Who are selected?

Priyanshu Mishra - 10 months, 1 week ago

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Use trigonometry as there is a right triangle formed assuming centet

Biswajit Barik - 10 months, 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 - 12 months ago

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What is your score in RMO?

Sayantan Saha - 11 months, 3 weeks ago

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I gave GMO

Racchit Jain - 11 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 - 11 months, 3 weeks ago

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@Racchit Jain Pls give for delhi region also.

Rajdeep Das - 11 months, 3 weeks ago

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@Rajdeep Das Is RMO DELHI result out?

Priyanshu Mishra - 11 months, 3 weeks ago

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@Priyanshu Mishra Yes

Rajdeep Das - 11 months, 3 weeks ago

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@Rajdeep Das At which website?

Priyanshu Mishra - 11 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 - 11 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 - 11 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 - 11 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 - 1 year ago

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this is north zone's (Delhi) problem

Sayantan Saha - 1 year ago

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could u do it?

Rajdeep Das - 1 year ago

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@Rajdeep Das Then pls post the solution

Rajdeep Das - 1 year ago

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@Rajdeep Das Show that AB is independent of the choice of point P

Racchit Jain - 1 year ago

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@Rajdeep Das Try using sine rule

Racchit Jain - 1 year ago

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@Racchit Jain It looks like Power of a Point, but Extended Sine Rule definitely works.

Sharky Kesa - 1 year ago

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@Sharky Kesa please add the solution

Sayantan Saha - 1 year 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 - 1 year ago

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@Sharky Kesa yeah. I have done it

Sayantan Saha - 1 year 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 - 1 year ago

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Please someone post this years rmo question paper.

Harsh Shrivastava - 1 year ago

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The papers are uploaded on AoPS.

Brilliant Member - 1 year ago

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Please give me the link.Thanks.

Harsh Shrivastava - 1 year 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 - 1 year ago

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Can someone post this year's problems?

Sharky Kesa - 1 year ago

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Mumbai region paper was really easy

Racchit Jain - 1 year ago

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Could you post the question paper please?

Kush Singhal - 1 year ago

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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 - 1 year ago

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@Racchit Jain Post it on Slack.

Sharky Kesa - 1 year ago

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@Sharky Kesa Umm...how do I do that?

Racchit Jain - 1 year 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 - 1 year ago

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@Racchit Jain You can do it on the browser.

Sharky Kesa - 1 year ago

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@Sharky Kesa Can I mail it to you and then you can post it?

Racchit Jain - 1 year ago

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@Racchit Jain Sure. sharkesa@gmail.com

Sharky Kesa - 1 year ago

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@Sharky Kesa Please post the paper.

Harsh Shrivastava - 1 year ago

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@Sharky Kesa I have sent it to you plz check

Racchit Jain - 1 year ago

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Did anyone give RMO from north zone?

Rajdeep Das - 1 year ago

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Uttar Pradesh - Me

Rishik Jain - 1 year ago

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At which center?

Rajdeep Das - 1 year ago

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@Rajdeep Das Meerut

Rishik Jain - 1 year ago

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Best of luck everyone

Rajdeep Das - 1 year 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 - 1 year ago

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Please post again as you cannot edit that.

Priyanshu Mishra - 1 year ago

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Sure I can! Mod powers! :P

Sharky Kesa - 1 year ago

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@Sharky Kesa But how you edited that?

Priyanshu Mishra - 1 year ago

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@Priyanshu Mishra With great skill (and a large screen)!

Sharky Kesa - 1 year ago

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@Sharky Kesa What is that skill!?

Priyanshu Mishra - 1 year ago

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@Priyanshu Mishra Big iMac skills! :P :P

Sharky Kesa - 1 year 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 - 1 year 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 - 1 year 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 - 1 year 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 - 1 year ago

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Hint : Use the fact that if a+b+c = 0 then \(a^3+b^3+c^3 = 3abc\)

Harsh Shrivastava - 1 year ago

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Good choice from MATHEMATICAL OLYMPIAD TREASURES.

Priyanshu Mishra - 1 year ago

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

As \(k\) is integer \(2k-2\) will also be an integer.

Akshat Sharda - 1 year ago

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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 - 1 year ago

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Firstly, your final statement is incorrect. Secondly, you have put no working. Sorry, but this is a null solution.

Sharky Kesa - 1 year ago

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

Please post difficult ones.

Priyanshu Mishra - 1 year ago

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@Priyanshu Mishra OK, sorry! I'll post IMO level probs next time.

Sharky Kesa - 1 year 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 - 1 year ago

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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 - 1 year ago

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I have a solution using trivial inequalities.

Harsh Shrivastava - 1 year ago

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@Harsh Shrivastava QM-AM is trivial.

Sharky Kesa - 1 year ago

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@Sharky Kesa Alright.Post some problems.

Harsh Shrivastava - 1 year ago

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@Sharky Kesa Yes after QM-AM the result directly follows.

Brilliant Member - 1 year ago

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I am also giving the RMO this year, please help me out 😀

Alan Joel - 1 year ago

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What is circumdiameter?

Harsh Shrivastava - 1 year ago

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sorry it is circumcenter.

Brilliant Member - 1 year ago

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Sorry guys, I wasn't active on brilliant (and might not be for a period of time) Best luck for your rmos

Nihar Mahajan - 1 year ago

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Actually RMO happened today for most of the regions. Some are on 16th.

Brilliant Member - 1 year ago

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@Brilliant Member yep, i gave it today :)

Nihar Mahajan - 1 year ago

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@Nihar Mahajan Please most the paper.

Harsh Shrivastava - 1 year ago

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@Harsh Shrivastava How to most a paper? 😂😂 XD

Nihar Mahajan - 1 year ago

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@Nihar Mahajan Lol I meant post.Please post your paper.

Harsh Shrivastava - 1 year ago

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@Nihar Mahajan How was it?

Brilliant Member - 1 year ago

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@Brilliant Member How was ur rmo?how many did get correct?

Naitik Sanghavi - 1 year ago

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@Naitik Sanghavi Please most the paper.

Harsh Shrivastava - 1 year ago

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@Naitik Sanghavi Mine is on 16th. How was your paper?

Brilliant Member - 1 year ago

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@Brilliant Member 3-4 correct ,This time 2 question were very easy, so may be cutoff will go high!

Naitik Sanghavi - 1 year ago

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@Naitik Sanghavi From which region did u give RMO?

Rajdeep Das - 1 year ago

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@Rajdeep Das Gujarat

Naitik Sanghavi - 1 year ago

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@Brilliant Member pretty good, better than last time's

Nihar Mahajan - 1 year ago

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Thanks. But I left Olympiad mathematics forever.

Swapnil Das - 1 year ago

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Are you ok? Say no

Nihar Mahajan - 1 year 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 - 1 year ago

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WHATT!!!!!!!!!!!!!

Harsh Shrivastava - 1 year ago

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@Harsh Shrivastava That's true.

Swapnil Das - 1 year ago

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Here's the link to last year's RMO board

Harsh Shrivastava - 1 year 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 - 1 year ago

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It is not over in many regions, mine is on 16th.

Harsh Shrivastava - 1 year ago

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Even mine is on 16th

Alan Joel - 1 year ago

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How was your rmo :)?

Nihar Mahajan - 12 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 - 12 months ago

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@Harsh Shrivastava Your Olympiad maths journey is not over. Congo :)

Nihar Mahajan - 10 months, 1 week ago

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

How was KVPY?

Harsh Shrivastava - 10 months, 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 - 10 months, 1 week ago

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@Harsh Shrivastava Do you get selected in RMO?

I got selected.

Priyanshu Mishra - 10 months, 1 week ago

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@Priyanshu Mishra Yes.Let's start the INMO Board!!

Harsh Shrivastava - 10 months, 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 - 10 months, 1 week ago

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@Priyanshu Mishra I think you should post a new note because that thread has died.

Harsh Shrivastava - 10 months, 1 week ago

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@Harsh Shrivastava Ok i will post a fresh one.

Priyanshu Mishra - 10 months, 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 - 12 months ago

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@Nihar Mahajan Did you give from Mumbai region?

Racchit Jain - 11 months, 3 weeks ago

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@Nihar Mahajan Not sure about that rocking part :(

Harsh Shrivastava - 12 months ago

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So wait upto that and then solve INMO problems.

Priyanshu Mishra - 1 year ago

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I have RMO on 23rd.

Ayush Pattnayak - 1 year ago

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