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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
7 months, 4 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° · 7 months, 3 weeks ago

can u post the solution for the 2nd part of problem 1 · 7 months, 2 weeks ago

I will try to solve it and if I succeed ,I'll post the solution. · 7 months, 2 weeks 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] · 7 months, 2 weeks ago

sorry https://postimg.org/image/ylrffpqh1/ https://postimg.org/image/e0cnnt8w5/ open the links · 7 months, 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!! · 7 months, 2 weeks ago

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. · 1 month, 4 weeks ago

Oh that's awesome use of trignometery! · 7 months, 2 weeks ago

I can't edit my comment, its "AHCM". · 7 months, 3 weeks ago

Brilliant staff are working on this issue. · 7 months, 3 weeks ago

Hey Sharky post some problems (RMO level.) · 7 months, 3 weeks 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 $$\angle DEP = \angle PFD$$. · 7 months, 3 weeks 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. · 7 months, 3 weeks ago

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

=> $$\Delta EBD\sim \Delta FCD$$.

=> $$\angle BED$$ = $$\angle CFD$$ =>$$\angle DEP = \angle PFD$$.

$$KIPKIG.$$ · 1 month, 3 weeks ago

· 1 month, 3 weeks ago

In $$\Delta ABC$$, O is the circumcenter and H is the orthocenter. Prove that $$AH^2+BC^2=4AO^2$$. · 7 months, 3 weeks ago

Draw $$OD\perp BC$$.

Just $$Pythagoras$$ then, $$BD^{2}$$ $$+$$ $$OD^{2}$$ = $$BO^{2}$$

=> $$4$$ $$BD^{2}$$ $$+$$ $$4$$ $$OD^{2}$$ = $$4$$ $$BO^{2}$$

=> $$[2BD]^{2}$$ $$+$$ $$[2OD]^{2}$$ = $$4$$ $$BO^{2}$$

=> $$BC^{2}$$ + $$AH^{2}$$ = $$4$$ $$AO^{2}$$. · 1 month, 4 weeks ago

Can you post the solution please ? · 7 months, 2 weeks ago

i got it , its easy https://postimg.org/image/5d1sm6hm7/ · 7 months, 2 weeks ago

How did you get AH^2 = 2RcosA ? · 7 months, 2 weeks ago

Its an identity. · 7 months, 2 weeks ago

Its AH* · 7 months, 2 weeks ago

Oh, I knew that but I forgot lol · 7 months, 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. · 7 months, 4 weeks ago

Yes you are right. Thanks for pointing it out. · 7 months, 4 weeks ago

All INMO participants,please share ur marks. · 3 months, 3 weeks ago

70-80. · 3 months, 3 weeks ago

why dont u be active in slack? · 3 months, 3 weeks ago

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

I have a lot of stuffs for FIITJEE. I do that only. · 3 months, 3 weeks ago

Which centre? · 3 months, 2 weeks ago

oh...ok I am very sorry for disturbing u. · 3 months, 3 weeks ago

Its true. Kuch mazaa nahi aata slack chat pe. · 3 months, 3 weeks ago

Its not chatting.I invited bcoz ur an INMO qualifier and u can help us solve problems that are posted. · 3 months, 3 weeks ago

Initially i am only RMO qualifier. I am not INMO qualifier till the result is declared. · 3 months, 3 weeks ago

do u want to join my INMO team? · 3 months, 3 weeks ago

@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. · 4 months, 1 week ago

Me too.....ayushpattnayak2001@gmail.com · 2 months, 3 weeks ago

Mine is rajdeep.ind24@gmail.com · 4 months, 1 week ago

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 · 4 months, 1 week ago

I'm in, here's my email id alanj.33@cloud.com · 4 months, 1 week ago

I meant icloud* there · 4 months, 1 week ago

you can check ur mail now · 4 months, 1 week ago

Hello everybody,

RMO results are out.

Who are selected? · 5 months, 2 weeks ago

Use trigonometry as there is a right triangle formed assuming centet · 5 months, 2 weeks 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 · 7 months, 1 week ago

What is your score in RMO? · 6 months, 4 weeks ago

I gave GMO · 6 months, 4 weeks ago

However, I know marks of some of my friends from different regions, which region are you asking for? · 6 months, 4 weeks ago

Pls give for delhi region also. · 6 months, 4 weeks ago

Is RMO DELHI result out? · 6 months, 4 weeks ago

Yes · 6 months, 4 weeks ago

At which website? · 6 months, 4 weeks ago

I am from WB region. I want to know how high the scores of rmo had gone in Delhi this year. · 6 months, 4 weeks ago

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 · 6 months, 4 weeks ago

Only 35. I don't think so.

At which website is the result of RMO declared? · 6 months, 3 weeks 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. · 7 months, 2 weeks ago

this is north zone's (Delhi) problem · 7 months, 2 weeks ago

could u do it? · 7 months, 2 weeks ago

Then pls post the solution · 7 months, 2 weeks ago

Show that AB is independent of the choice of point P · 7 months, 2 weeks ago

Try using sine rule · 7 months, 2 weeks ago

It looks like Power of a Point, but Extended Sine Rule definitely works. · 7 months, 2 weeks ago

I won't give solution but the crux of this proof is to show that $$AB$$ is constant, irrespective of where $$P$$ is. · 7 months, 2 weeks ago

yeah. I have done it · 7 months, 2 weeks 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. · 7 months, 2 weeks ago

Please someone post this years rmo question paper. · 7 months, 2 weeks ago

The papers are uploaded on AoPS. · 7 months, 2 weeks ago

I have posted Gujarat rmo paper here,https://brilliant.org/discussions/thread/rmo-2016-gujarat-region/?ref_id=1272714 · 7 months, 2 weeks ago

Can someone post this year's problems? · 7 months, 2 weeks ago

Mumbai region paper was really easy · 7 months, 2 weeks ago

Could you post the question paper please? · 7 months, 2 weeks 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? · 7 months, 2 weeks ago

Post it on Slack. · 7 months, 2 weeks ago

Umm...how do I do that? · 7 months, 2 weeks ago

It's asking me too get the app can't I do it using the browser only? · 7 months, 2 weeks ago

You can do it on the browser. · 7 months, 2 weeks ago

Can I mail it to you and then you can post it? · 7 months, 2 weeks ago

Sure. sharkesa@gmail.com · 7 months, 2 weeks ago

Please post the paper. · 7 months, 2 weeks ago

I have sent it to you plz check · 7 months, 2 weeks ago

Did anyone give RMO from north zone? · 7 months, 2 weeks ago

Uttar Pradesh - Me · 7 months, 2 weeks ago

At which center? · 7 months, 2 weeks ago

Meerut · 7 months, 2 weeks ago

Best of luck everyone · 7 months, 3 weeks 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

$\displaystyle \sum_{i=1}^{\left \lfloor \frac{k}{2} \right \rfloor + 1} \left ( \left \lfloor \dfrac {n_i}{2} \right \rfloor + 1 \right ) .$ · 7 months, 3 weeks ago

Please post again as you cannot edit that. · 7 months, 3 weeks ago

Sure I can! Mod powers! :P · 7 months, 3 weeks ago

But how you edited that? · 7 months, 3 weeks ago

With great skill (and a large screen)! · 7 months, 3 weeks ago

What is that skill!? · 7 months, 3 weeks ago

Big iMac skills! :P :P · 7 months, 3 weeks 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. · 7 months, 3 weeks ago

Determine all positive triplets of integers such that

$$\large\ {(x+1)}^{y+1} + 1 = {(x+2)}^{z+1}.$$ · 7 months, 3 weeks 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. · 7 months, 3 weeks 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. · 7 months, 3 weeks ago

Hint : Use the fact that if a+b+c = 0 then $$a^3+b^3+c^3 = 3abc$$ · 7 months, 3 weeks ago

Good choice from MATHEMATICAL OLYMPIAD TREASURES. · 7 months, 3 weeks 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. · 7 months, 3 weeks 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. · 7 months, 3 weeks ago

Firstly, your final statement is incorrect. Secondly, you have put no working. Sorry, but this is a null solution. · 7 months, 3 weeks ago

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

Please post difficult ones. · 7 months, 3 weeks ago

OK, sorry! I'll post IMO level probs next time. · 7 months, 3 weeks 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. · 7 months, 3 weeks 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. · 7 months, 3 weeks ago

I have a solution using trivial inequalities. · 7 months, 3 weeks ago

QM-AM is trivial. · 7 months, 3 weeks ago

Alright.Post some problems. · 7 months, 3 weeks ago

Yes after QM-AM the result directly follows. · 7 months, 3 weeks ago

I am also giving the RMO this year, please help me out 😀 · 7 months, 4 weeks ago

What is circumdiameter? · 7 months, 4 weeks ago

sorry it is circumcenter. · 7 months, 4 weeks ago

Sorry guys, I wasn't active on brilliant (and might not be for a period of time) Best luck for your rmos · 7 months, 2 weeks ago

Actually RMO happened today for most of the regions. Some are on 16th. · 7 months, 2 weeks ago

yep, i gave it today :) · 7 months, 2 weeks ago

Please most the paper. · 7 months, 2 weeks ago

How to most a paper? 😂😂 XD · 7 months, 2 weeks ago

Lol I meant post.Please post your paper. · 7 months, 2 weeks ago

How was it? · 7 months, 2 weeks ago

How was ur rmo?how many did get correct? · 7 months, 2 weeks ago

Please most the paper. · 7 months, 2 weeks ago

Mine is on 16th. How was your paper? · 7 months, 2 weeks ago

3-4 correct ,This time 2 question were very easy, so may be cutoff will go high! · 7 months, 2 weeks ago

From which region did u give RMO? · 7 months, 2 weeks ago

Gujarat · 7 months, 2 weeks ago

pretty good, better than last time's · 7 months, 2 weeks ago

Thanks. But I left Olympiad mathematics forever. · 7 months, 4 weeks ago

Are you ok? Say no · 7 months, 2 weeks 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. · 7 months, 2 weeks ago

WHATT!!!!!!!!!!!!! · 7 months, 4 weeks ago

That's true. · 7 months, 4 weeks ago

Here's the link to last year's RMO board · 7 months, 4 weeks 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.

https://brilliant.org/discussions/thread/inmo-2017-board/. · 7 months, 2 weeks ago

It is not over in many regions, mine is on 16th. · 7 months, 2 weeks ago

Even mine is on 16th · 7 months, 2 weeks ago

How was your rmo :)? · 7 months, 1 week 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? · 7 months, 1 week ago

Your Olympiad maths journey is not over. Congo :) · 5 months, 2 weeks ago

Also,how's ya' iit prep going on?

How was KVPY? · 5 months, 2 weeks ago

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? · 5 months, 2 weeks ago

Do you get selected in RMO?

I got selected. · 5 months, 2 weeks ago

Yes.Let's start the INMO Board!! · 5 months, 2 weeks ago

Oh congratulations.

I have already started that. Check here.

https://brilliant.org/discussions/thread/inmo-2017-board/?ref_id=1273098 · 5 months, 2 weeks ago

I think you should post a new note because that thread has died. · 5 months, 2 weeks ago

Ok i will post a fresh one. · 5 months, 2 weeks 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 😀😀 · 7 months, 1 week ago

Did you give from Mumbai region? · 6 months, 4 weeks ago

Not sure about that rocking part :( · 7 months, 1 week ago

So wait upto that and then solve INMO problems. · 7 months, 2 weeks ago