# 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 A Former Brilliant Member
3 years, 1 month ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

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 years, 1 month ago

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

- 3 years, 1 month ago

Brilliant staff are working on this issue.

- 3 years, 1 month ago

Hey Sharky post some problems (RMO level.)

- 3 years, 1 month ago

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

- 3 years, 1 month ago

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

- 3 years, 1 month 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]

- 3 years, 1 month ago

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

- 3 years, 1 month ago

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

- 3 years, 1 month ago

Oh that's awesome use of trignometery!

- 3 years, 1 month 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.

- 2 years, 7 months ago

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

- 2 years, 5 months ago

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 $AH = 2OD$.

- 2 years, 5 months 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 years, 1 month 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 years, 1 month ago - 2 years, 7 months 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 $\dfrac{BD}{CD}$ = $\dfrac{AB}{AC}$ => $\dfrac{BE}{CF}$ = $\dfrac{BD}{CD}$.

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

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

$KIPKIG.$

- 2 years, 7 months ago

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

- 2 years, 5 months ago

yes ..like geogebra

- 2 years, 5 months 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 years, 1 month ago

Yes you are right. Thanks for pointing it out.

- 3 years, 1 month ago

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

- 3 years, 1 month ago

Draw $OD\perp BC$.

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

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

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

=> $BC^{2}$ + $AH^{2}$ = $4$ $AO^{2}$.

- 2 years, 7 months ago

Can you post the solution please ?

- 3 years, 1 month ago

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

- 3 years, 1 month ago

How did you get AH^2 = 2RcosA ?

- 3 years, 1 month ago

Its an identity.

- 3 years, 1 month ago

Its AH*

- 3 years, 1 month ago

Oh, I knew that but I forgot lol

- 3 years, 1 month ago

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

- 3 years, 1 month ago

- 3 years, 1 month ago

Thanks. But I left Olympiad mathematics forever.

- 3 years, 1 month ago

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

- 3 years, 1 month ago

That's true.

- 3 years, 1 month ago

Are you ok? Say no

- 3 years, 1 month 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.

- 3 years, 1 month ago

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

- 3 years, 1 month ago

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

- 3 years, 1 month ago

yep, i gave it today :)

- 3 years, 1 month ago

How was it?

- 3 years, 1 month ago

pretty good, better than last time's

- 3 years, 1 month ago

How was ur rmo?how many did get correct?

- 3 years, 1 month ago

Mine is on 16th. How was your paper?

- 3 years, 1 month ago

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

- 3 years, 1 month ago

From which region did u give RMO?

- 3 years, 1 month ago

Gujarat

- 3 years, 1 month ago

- 3 years, 1 month ago

- 3 years, 1 month ago

How to most a paper? 😂😂 XD

- 3 years, 1 month ago

- 3 years, 1 month ago

What is circumdiameter?

- 3 years, 1 month ago

sorry it is circumcenter.

- 3 years, 1 month ago

- 3 years, 1 month 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 years, 1 month 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 years, 1 month ago

I have a solution using trivial inequalities.

- 3 years, 1 month ago

QM-AM is trivial.

- 3 years, 1 month ago

Yes after QM-AM the result directly follows.

- 3 years, 1 month ago

Alright.Post some problems.

- 3 years, 1 month ago

3) Let $k$ be an integer and let

$n=\sqrt{k+\sqrt{k^2-1}} + \sqrt{k-\sqrt{k^2-1}}+1$

Prove $n^3 - 3n^2$ is an integer.

- 3 years, 1 month ago

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

- 3 years, 1 month ago

Good choice from MATHEMATICAL OLYMPIAD TREASURES.

- 3 years, 1 month ago

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

As $k$ is integer $2k-2$ will also be an integer.

- 3 years, 1 month ago

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

- 3 years, 1 month ago

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

- 3 years, 1 month ago

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

- 3 years, 1 month 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 years, 1 month 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 years, 1 month ago

Determine all positive triplets of integers such that

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

- 3 years, 1 month 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 years, 1 month 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 years, 1 month ago

Please post again as you cannot edit that.

- 3 years, 1 month ago

Sure I can! Mod powers! :P

- 3 years, 1 month ago

But how you edited that?

- 3 years, 1 month ago

With great skill (and a large screen)!

- 3 years, 1 month ago

What is that skill!?

- 3 years, 1 month ago

Big iMac skills! :P :P

- 3 years, 1 month ago

Best of luck everyone

- 3 years, 1 month ago

Did anyone give RMO from north zone?

- 3 years, 1 month ago

- 3 years, 1 month ago

At which center?

- 3 years, 1 month ago

Meerut

- 3 years, 1 month ago

Mumbai region paper was really easy

- 3 years, 1 month ago

Could you post the question paper please?

- 3 years, 1 month 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?

- 3 years, 1 month ago

Post it on Slack.

- 3 years, 1 month ago

Umm...how do I do that?

- 3 years, 1 month ago

It's asking me too get the app can't I do it using the browser only?

- 3 years, 1 month ago

You can do it on the browser.

- 3 years, 1 month ago

Can I mail it to you and then you can post it?

- 3 years, 1 month ago

Sure. sharkesa@gmail.com

- 3 years, 1 month ago

I have sent it to you plz check

- 3 years, 1 month ago

- 3 years, 1 month ago

Can someone post this year's problems?

- 3 years, 1 month ago

Please someone post this years rmo question paper.

- 3 years, 1 month ago

The papers are uploaded on AoPS.

- 3 years, 1 month ago

- 3 years, 1 month ago

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

- 3 years, 1 month 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.

- 3 years, 1 month ago

this is north zone's (Delhi) problem

- 3 years, 1 month ago

could u do it?

- 3 years, 1 month ago

Then pls post the solution

- 3 years, 1 month ago

Show that AB is independent of the choice of point P

- 3 years, 1 month ago

Try using sine rule

- 3 years, 1 month ago

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

- 3 years, 1 month ago

- 3 years, 1 month ago

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

- 3 years, 1 month ago

yeah. I have done it

- 3 years, 1 month 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.

- 3 years, 1 month 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

- 3 years ago

What is your score in RMO?

- 3 years ago

I gave GMO

- 3 years ago

However, I know marks of some of my friends from different regions, which region are you asking for?

- 3 years ago

I am from WB region. I want to know how high the scores of rmo had gone in Delhi this year.

- 3 years 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

- 3 years ago

Only 35. I don't think so.

At which website is the result of RMO declared?

- 3 years ago

Pls give for delhi region also.

- 3 years ago

Is RMO DELHI result out?

- 3 years ago

Yes

- 3 years ago

At which website?

- 3 years ago

Use trigonometry as there is a right triangle formed assuming centet

- 2 years, 11 months ago

Hello everybody,

RMO results are out.

Who are selected?

- 2 years, 11 months 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.

- 2 years, 10 months ago

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

- 2 years, 10 months ago

I meant icloud* there

- 2 years, 10 months ago

you can check ur mail now

- 2 years, 10 months ago

Mine is rajdeep.ind24@gmail.com

- 2 years, 10 months 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

- 2 years, 10 months ago

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

- 2 years, 8 months ago

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

- 2 years, 1 month ago

All INMO participants,please share ur marks.

- 2 years, 9 months ago

do u want to join my INMO team?

- 2 years, 9 months ago

70-80.

- 2 years, 9 months ago

why dont u be active in slack?

- 2 years, 9 months ago

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

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

- 2 years, 9 months ago

oh...ok I am very sorry for disturbing u.

- 2 years, 9 months ago

Its true. Kuch mazaa nahi aata slack chat pe.

- 2 years, 9 months ago

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

- 2 years, 9 months ago

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

- 2 years, 9 months ago

Which centre?

- 2 years, 9 months ago

@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!!

- 2 years, 1 month ago

u just need some angle chasing and similarity for RMO.

- 2 years, 1 month ago

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!

- 2 years, 1 month 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.

- 3 years, 1 month ago

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

- 3 years, 1 month ago

Even mine is on 16th

- 3 years, 1 month ago

So wait upto that and then solve INMO problems.

- 3 years, 1 month ago

- 3 years 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.

- 3 years 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 years ago

Not sure about that rocking part :(

- 3 years ago

Did you give from Mumbai region?

- 3 years ago

- 2 years, 11 months 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?

- 2 years, 11 months ago

Do you get selected in RMO?

I got selected.

- 2 years, 11 months ago

Yes.Let's start the INMO Board!!

- 2 years, 11 months ago

Oh congratulations.

I have already started that. Check here.

- 2 years, 11 months ago

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

- 2 years, 11 months ago

Ok i will post a fresh one.

- 2 years, 11 months ago

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

How was KVPY?

- 2 years, 11 months ago

I have RMO on 23rd.

- 3 years, 1 month ago