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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.

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 weeks, 5 days 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° · 3 weeks, 2 days ago

can u post the solution for the 2nd part of problem 1 · 2 weeks, 1 day ago

I will try to solve it and if I succeed ,I'll post the solution. · 2 weeks, 1 day ago

i got it here is the solution [url=https://postimg.org/image/e0cnnt8w5/][img]https://s18.postimg.org/e0cnnt8w5/IMG_0100.jpg[/img][/url] · 2 weeks ago

sorry https://postimg.org/image/ylrffpqh1/ https://postimg.org/image/e0cnnt8w5/ open the links · 2 weeks ago

the 1 st part can be solved much much easily AO=AH R=2RcosA 2cosA=1 cosA=1/2 A=60 done!! · 2 weeks, 1 day ago

Oh that's awesome use of trignometery! · 2 weeks, 1 day ago

I can't edit my comment, its "AHCM". · 3 weeks, 2 days ago

Brilliant staff are working on this issue. · 3 weeks, 2 days ago

Hey Sharky post some problems (RMO level.) · 3 weeks, 2 days ago

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 $$\angle DEP = \angle PFD$$. · 3 weeks, 1 day ago

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. · 3 weeks, 1 day ago

In $$\Delta ABC$$, O is the circumcenter and H is the orthocenter. Prove that $$AH^2+BC^2=4AO^2$$. · 3 weeks, 1 day ago

Can you post the solution please ? · 2 weeks ago

i got it , its easy https://postimg.org/image/5d1sm6hm7/ · 2 weeks ago

How did you get AH^2 = 2RcosA ? · 2 weeks ago

Its an identity. · 2 weeks ago

Its AH* · 2 weeks ago

Oh, I knew that but I forgot lol · 2 weeks ago

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. · 3 weeks, 3 days ago

Yes you are right. Thanks for pointing it out. · 3 weeks, 3 days ago

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 · 5 days, 4 hours ago

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.

I have RMO on 23rd. · 1 week, 4 days ago

It is not over in many regions, mine is on 16th. · 2 weeks ago

Even mine is on 16th · 2 weeks ago

How was your rmo :)? · 4 days ago

Well it could not have been any more 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? · 4 days ago

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 😀😀 · 3 days, 14 hours ago

Not sure about that rocking part :( · 3 days, 13 hours ago

So wait upto that and then solve INMO problems. · 1 week, 5 days ago

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. · 2 weeks, 1 day ago

this is north zone's (Delhi) problem · 2 weeks, 1 day ago

could u do it? · 2 weeks, 1 day ago

Then pls post the solution · 2 weeks, 1 day ago

Show that AB is independent of the choice of point P · 2 weeks, 1 day ago

Try using sine rule · 2 weeks, 1 day ago

It looks like Power of a Point, but Extended Sine Rule definitely works. · 2 weeks, 1 day ago

I won't give solution but the crux of this proof is to show that $$AB$$ is constant, irrespective of where $$P$$ is. · 2 weeks, 1 day ago

yeah. I have done it · 2 weeks, 1 day ago

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. · 2 weeks, 1 day ago

Please someone post this years rmo question paper. · 2 weeks, 1 day ago

The papers are uploaded on AoPS. · 2 weeks, 1 day ago

I have posted Gujarat rmo paper here,https://brilliant.org/discussions/thread/rmo-2016-gujarat-region/?ref_id=1272714 · 2 weeks, 1 day ago

Can someone post this year's problems? · 2 weeks, 1 day ago

Mumbai region paper was really easy · 2 weeks, 1 day ago

Could you post the question paper please? · 2 weeks, 1 day ago

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? · 2 weeks, 1 day ago

Post it on Slack. · 2 weeks, 1 day ago

Umm...how do I do that? · 2 weeks, 1 day ago

It's asking me too get the app can't I do it using the browser only? · 2 weeks, 1 day ago

You can do it on the browser. · 2 weeks, 1 day ago

Can I mail it to you and then you can post it? · 2 weeks, 1 day ago

Sure. sharkesa@gmail.com · 2 weeks, 1 day ago

Please post the paper. · 2 weeks, 1 day ago

I have sent it to you plz check · 2 weeks, 1 day ago

Did anyone give RMO from north zone? · 2 weeks, 2 days ago

Uttar Pradesh - Me · 2 weeks, 2 days ago

At which center? · 2 weeks, 1 day ago

Meerut · 2 weeks, 1 day ago

Best of luck everyone · 2 weeks, 3 days ago

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

$\displaystyle \sum_{i=1}^{\left \lfloor \frac{k}{2} \right \rfloor + 1} \left ( \left \lfloor \dfrac {n_i}{2} \right \rfloor + 1 \right ) .$ · 3 weeks, 1 day ago

Please post again as you cannot edit that. · 3 weeks, 1 day ago

Sure I can! Mod powers! :P · 3 weeks, 1 day ago

But how you edited that? · 3 weeks, 1 day ago

With great skill (and a large screen)! · 3 weeks, 1 day ago

What is that skill!? · 3 weeks, 1 day ago

Big iMac skills! :P :P · 3 weeks, 1 day ago

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

Find the largest real solution to this equation. · 3 weeks, 1 day ago

Determine all positive triplets of integers such that

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

@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. · 3 weeks, 1 day ago

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. · 3 weeks, 2 days ago

Hint : Use the fact that if a+b+c = 0 then $$a^3+b^3+c^3 = 3abc$$ · 3 weeks, 2 days ago

Good choice from MATHEMATICAL OLYMPIAD TREASURES. · 3 weeks, 2 days ago

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

As $$k$$ is integer $$2k-2$$ will also be an integer. · 3 weeks, 1 day ago

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. · 3 weeks, 1 day ago

Firstly, your final statement is incorrect. Secondly, you have put no working. Sorry, but this is a null solution. · 3 weeks, 1 day ago

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

Please post difficult ones. · 3 weeks, 1 day ago

OK, sorry! I'll post IMO level probs next time. · 3 weeks, 1 day ago

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. · 3 weeks, 2 days ago

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. · 3 weeks, 2 days ago

I have a solution using trivial inequalities. · 3 weeks, 2 days ago

QM-AM is trivial. · 3 weeks, 2 days ago

Alright.Post some problems. · 3 weeks, 2 days ago

Yes after QM-AM the result directly follows. · 3 weeks, 2 days ago

I am also giving the RMO this year, please help me out 😀 · 3 weeks, 4 days ago

What is circumdiameter? · 3 weeks, 5 days ago

sorry it is circumcenter. · 3 weeks, 5 days ago

Sorry guys, I wasn't active on brilliant (and might not be for a period of time) Best luck for your rmos · 2 weeks, 2 days ago

Actually RMO happened today for most of the regions. Some are on 16th. · 2 weeks, 2 days ago

yep, i gave it today :) · 2 weeks, 2 days ago

Please most the paper. · 2 weeks, 2 days ago

How to most a paper? 😂😂 XD · 2 weeks, 2 days ago

Lol I meant post.Please post your paper. · 2 weeks, 1 day ago

How was it? · 2 weeks, 2 days ago

How was ur rmo?how many did get correct? · 2 weeks, 2 days ago

Please most the paper. · 2 weeks, 2 days ago

Mine is on 16th. How was your paper? · 2 weeks, 2 days ago

3-4 correct ,This time 2 question were very easy, so may be cutoff will go high! · 2 weeks, 2 days ago

From which region did u give RMO? · 2 weeks, 2 days ago

Gujarat · 2 weeks, 2 days ago

pretty good, better than last time's · 2 weeks, 2 days ago

Thanks. But I left Olympiad mathematics forever. · 3 weeks, 5 days ago

Are you ok? Say no · 2 weeks, 2 days ago

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. · 2 weeks, 2 days ago

WHATT!!!!!!!!!!!!! · 3 weeks, 5 days ago

That's true. · 3 weeks, 5 days ago