RMO Online Mock Test - Results

So, RMO online mock test is completed.

Only 6 members had sent their solutions. So, here are the results.

So, Congratulations Nihar. You stood first.

Your prizes(ebooks) will be sent to your mail soon.

#RMO #RMOOnlineTest

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.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

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

Math	Appears 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}`	$\boxed{123}$

Comments

Names of the ebooks prizes ? ( I guess you won't mind to reveal :) ).

Venkata Karthik Bandaru - 5 years, 8 months ago

Hey Guys I will just give you an easy solution to problem 2.

It can be easily observed that $E$ is the midpoint of arc $AC$ containing $B$ . Let us drop a perpendicular $N$ from $E$ to on to $BC$ . According to Archimedes Broken Chord Theorem, $CN = BN + AB = BN + BF = FN$ . So, $N$ is midpoint of $CF$ . It implies that $EN$ is perpendicular bisector of $CF$ . Now, it is easy to prove that $DE$ is perpendicular bisector of $AC$ . As perpendicular bisectors of $CF$ and $AC$ i.e. $EN$ and $ED$ respectively meet at $E$ . So, $E$ is the circumcenter of $\Delta AFC$ .

Hence Proved.

Surya Prakash - 5 years, 8 months ago

It would be better if you even mention proof to Archimedes broken chord.

Surya Prakash - 5 years, 8 months ago

All I did was prove congruences and did a bit of angle chasing. My solution is longer but simpler.

Sharky Kesa - 5 years, 8 months ago

Can you post the solutions to the first and the second questions?@Surya Prakash

Adarsh Kumar - 5 years, 8 months ago

Yes I too want to see the solution for the second one.

A Former Brilliant Member - 5 years, 8 months ago

How many of the problems did you get?

Adarsh Kumar - 5 years, 8 months ago

@Adarsh Kumar – Tried the first 4 got 2.

A Former Brilliant Member - 5 years, 8 months ago

@Adarsh Kumar – I got all 6.

Sharky Kesa - 5 years, 8 months ago

Thank you everyone :) Also thanks @Surya Prakash for holding this contest. Hope to see more challenging upcoming contests :)

Nihar Mahajan - 5 years, 8 months ago

Congrats! Nihar Mahajan.You did best!

Siddharth Singh - 5 years, 8 months ago

@Nihar Mahajan @Sharky Kesa @Shivam Jadhav @Siddharth Singh @Svatejas Shivakumar @Sarthak Behera

Surya Prakash - 5 years, 8 months ago

Can we get our individual question marks and best solutions for each problem?

Sharky Kesa - 5 years, 8 months ago

Congratulations! @Nihar Mahajan

Swapnil Das - 5 years, 8 months ago

Congratulations Nihar!!

A Former Brilliant Member - 5 years, 8 months ago

Congrats nihar but want to ask you did you solve the paper in 3 hrs because I did .

Shivam Jadhav - 5 years, 8 months ago

I did the first 5 questions in 2 hours. The 6th question required time though...

Nihar Mahajan - 5 years, 8 months ago

Can you please post your solution for the second question?

A Former Brilliant Member - 5 years, 8 months ago

@A Former Brilliant Member – Sure.

Nihar Mahajan - 5 years, 8 months ago

@Nihar Mahajan – Can you post solution to 5,6 problem

Shivam Jadhav - 5 years, 8 months ago

@Shivam Jadhav – Q5)

Extend AI to meet the circumcircle of $\Delta ABC$ at $X$ , $BI$ at $Y$ and $CI$ at $Z$ . Note that $\angle BAX=\angle BCX$ and $\angle CAX = \angle CBX$ by cyclic bowties (Angles subtended by the same chord). Also, $\angle BAX=\angle CAX$ because $AX$ is the angle bisector of $\angle BAC$ . Therefore, $\angle CBX=\angle BCX$ , which implies $\Delta BXC$ is an isosceles triangle, which further implies $BX=CX$ . Now, we have to prove $BX=IX$ .

We have

$\begin{aligned} \angle XBI&=\angle XBC+\angle CBI\\ &=\angle XAC+\angle CBI\\ &=\angle IAC+\angle CBI\\ &=\dfrac {1}{2} \angle BAC + \dfrac {1}{2} \angle ABC \end{aligned}$

Also,

$\begin{aligned} \angle XIB&=\angle IAB+ \angle IBA \text{(}\angle XIB \text{ is the external angle of } \Delta AIB\text{)}\\ &=\dfrac{1}{2} \angle BAC+\dfrac{1}{2} \angle ABC \end{aligned}$

Thus, $\angle XBI=\angle XIB$ , which implies $\Delta XIB$ is isosceles, which implies $XI=XB$ . We already know that $XB=XC$ , so $X$ is the circumcentre of $\Delta IBC$ . By definition, point $X$ is on the circumcircle of $\Delta ABC$ . Therefore, the circumcentre of $\Delta IBC$ lies on the circumcircle of $\Delta ABC$ . Similarly, it can be proven that the circumcentre ( $Y$ ) of $\Delta ICA$ lies on the circumcircle of $\Delta ABC$ and the circumcentre ( $Z$ ) of $\Delta IAB$ lies on the circumcircle of $\Delta ABC$ .

Sharky Kesa - 5 years, 8 months ago

@Sharky Kesa – I would not recommend my solution for Q6 since it is (a) 4 pages long and (b) Proves a massive generalisation.

Sharky Kesa - 5 years, 8 months ago

@Sharky Kesa – What!Really?!My solution is a page long!

Adarsh Kumar - 5 years, 8 months ago

@Adarsh Kumar – Q6, Q3

Adarsh Kumar - 5 years, 8 months ago

@Adarsh Kumar – @Shivam Jadhavcould you post the solution to the first one?

Adarsh Kumar - 5 years, 8 months ago

@Adarsh Kumar – $ab(a+b)+ac(a+c)+bc(b+c)= \dfrac{ab(a+b)+ac(a+c)+bc(b+c)+ab(a+b)+ac(a+c)+bc(b+c)}{2}=\dfrac{a^2(b+c)+b^2(a+c)+c^2(a+b)+(a^2b+b^2c+a^2c)+(ab^2+bc^2+ac^2)}{2}$

By AM-GM inequality, $ab^2+bc^2+ac^2 \ge 3abc$

Therefore,

$\dfrac{a^2(b+c)+b^2(a+c)+c^2(a+b)+(a^2b+b^2c+a^2c)+(ab^2+bc^2+ac^2)}{2} \ge \dfrac{a^2(b+c)+b^2(a+c)+c^2(a+b)+(a^2b+b^2c+a^2c)+3abc}{2}=\dfrac{a^2(b+c)+b^2(a+c)+c^2(a+b)+ab(a+c)+bc(a+b)+ac(b+c)}{2}$

Grouping and again applying AM-GM inequality, (Applying AM-GM inequality on each term and adding all the terms)

$\dfrac{(a^2(b+c)+ab(a+c))}{2}+\dfrac{(b^2(a+c)+bc(a+b))+}{2}+\dfrac{(c^2(a+b)+ac(b+c))}{2} \ge \sqrt{a^3b(b+c)(a+c)}+\sqrt{b^3c(a+c)(a+b)}+\sqrt{c^3a(a+b)(a+c)}=\displaystyle\sum_{cyc}ab\sqrt{\dfrac{a}{b}(b+c)(c+a)}.$ Equality occurs when $a=b=c$

Hence proved.

A Former Brilliant Member - 5 years, 8 months ago

@A Former Brilliant Member – Woah!That is a big solution!Could you write the motivation please?

Adarsh Kumar - 5 years, 8 months ago

Congratulations @Nihar Mahajan !

Adarsh Kumar - 5 years, 8 months ago

@Surya Prakash Thank you very much for holding the content. When will the second one start?

A Former Brilliant Member - 5 years, 8 months ago

I can't say when because I am somewhat busy with preparation for NSEP.

Surya Prakash - 5 years, 8 months ago

I also want to participate insuch contests . Can any one suggest me the pathway and dates for future such contests

Aakash Khandelwal - 5 years, 8 months ago

Thanks! to Surya Prakash for this opportunity.

Siddharth Singh - 5 years, 8 months ago

Hey guys. The ebooks i.e. prizes are sent to the participants. Please check your mails.

Surya Prakash - 5 years, 8 months ago

When was it held?

Anandhu Raj - 5 years, 7 months ago

@Surya Prakash Could u please conduct an INMO mock test similar to this ?

Shrihari B - 5 years, 5 months ago

Math	Appears 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}`	$\boxed{123}$