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 \(\angle A=60^\circ\).

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## Comments

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TopNewestSolution1)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°

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

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Brilliant staff are working on this issue.

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

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

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@Svatejas Shivakumar 's question :

See an even more efficient use of \(trigonometry\). 1st Part of\(1.\) 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.

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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 \(\angle DEP = \angle PFD\).

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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.

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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 \(\dfrac{BD}{CD}\) = \(\dfrac{AB}{AC}\) => \(\dfrac{BE}{CF}\) = \(\dfrac{BD}{CD}\).

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

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

\(KIPKIG.\)

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

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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.

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Yes you are right. Thanks for pointing it out.

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

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Draw \(OD\perp BC\).

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

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

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

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

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Can you post the solution please ?

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

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

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@Vaibhav Prasad @Kalash Verma @Nihar Mahajan @Adarsh Kumar @Akshat Sharda @AkshayYadav @Swapnil Das @Rajdeep Dhingra @Anik Mandal @Lakshya Sinha @Abhay Kumar @Priyanshu Mishra @Dev Sharma @Sharky Kesa and everyone!

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Thanks. But I left Olympiad mathematics forever.

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

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Are you ok? Say no

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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.

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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

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

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What is circumdiameter?

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sorry it is circumcenter.

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

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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.

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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.

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I have a solution using trivial inequalities.

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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.

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

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Good choice from MATHEMATICAL OLYMPIAD TREASURES.

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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.

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

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Please post difficult ones.

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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.

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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.

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Determine all positive triplets of integers such that

\(\large\ {(x+1)}^{y+1} + 1 = {(x+2)}^{z+1}.\)

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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.

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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

\[\displaystyle \sum_{i=1}^{\left \lfloor \frac{k}{2} \right \rfloor + 1} \left ( \left \lfloor \dfrac {n_i}{2} \right \rfloor + 1 \right ) . \]

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Please post again as you cannot edit that.

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Sure I can! Mod powers! :P

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Best of luck everyone

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Did anyone give RMO from north zone?

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Uttar Pradesh - Me

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At which center?

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Mumbai region paper was really easy

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Could you post the question paper please?

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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?

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Can someone post this year's problems?

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Please someone post this years rmo question paper.

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The papers are uploaded on AoPS.

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Please give me the link.Thanks.

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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.

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this is north zone's (Delhi) problem

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could u do it?

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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

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What is your score in RMO?

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I gave GMO

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At which website is the result of RMO declared?

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Use trigonometry as there is a right triangle formed assuming centet

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Hello everybody,

RMO results are out.

Who are selected?

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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.

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

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I meant icloud* there

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Mine is rajdeep.ind24@gmail.com

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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

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

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

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All INMO participants,please share ur marks.

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do u want to join my INMO team?

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70-80.

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why dont u be active in slack?

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

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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!!

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u just need some angle chasing and similarity for RMO.

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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!

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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/.

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

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Even mine is on 16th

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So wait upto that and then solve INMO problems.

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How was your rmo :)?

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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?

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Now it's ok, Harsh, I know you are gonna rock in other exams 😀😀

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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?

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I got selected.

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I have already started that. Check here.

https://brilliant.org/discussions/thread/inmo-2017-board/?ref_id=1273098

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How was KVPY?

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I have RMO on 23rd.

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