Annoying algebra in today's pre-RMO !

Hello !

Today was my Pre-RMO . I was stuck in the following question please help .

$\sin^{-1} x + \sin^{-1} y + \sin^{-1} z = \dfrac{3 \pi}{2}$

Then find the value of :

$x^9 + y^9 + z^9 + \dfrac{1}{x^9y^9z^9}$

Well , I had got the following result from this : ( Might be wrong )

$x + y + z = -1$

but I couldn't manage to do it any further . Please help .

#Algebra #OlympiadMath #RMO #HelpMe! #Proofs #ExamStrategies #Advice #Competitions #MathCompetitions #MathProblem #Math #Opinions

Easy Math Editor

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Comments

The range of $\sin^{-1}$ is $[-\tfrac12\pi,\tfrac12\pi]$ . Since $\sin^{-1}x+\sin^{-1}y+\sin^{-1}z = \tfrac32\pi$ , we know the values of $\sin^{-1}x,\sin^{-1}y,\sin^{-1}z$ straight away (what three numbers between $-a$ and $a$ add to $3a$ ?). Thus we know $x,y,z$ .

Mark Hennings - 7 years, 8 months ago

Thank You !

Priyansh Sangule - 7 years, 8 months ago

Very simple one! As, $\sin^{-1} x \to [\frac{- \pi}{2}, \frac{\pi}{2}]$

So, it leads us to the conclusion that $x=1, y=1, z=1$

Then, required value becomes $\boxed{4}$

Advitiya Brijesh - 7 years, 8 months ago

i thought same . quite easy.

umang garg - 7 years, 6 months ago

max value of the following equation is 3pi/2 and hence each angle should be pi/2. if angle is pi/2 then x=y=z=1 then 1 + 1 + 1 + 1 = 4. That's it

Vaibhav Reddy - 7 years, 8 months ago

What is Pre-RMO? Is the full paper available somewhere?

Thanks!

Pranav Arora - 7 years, 8 months ago

Well this time , the government has allowed every state to set their own papers for pre-RMO . That is the qualification stage for RMO level 1 .

Priyansh Sangule - 7 years, 8 months ago

In Telangana region,there is no pre-RMO....lucky for me I can directly write RMO paper without qualifying.... BTW....The RMO exam is not so easy as pre-RMO....I was dying to solve some of them...

Vaibhav Reddy - 7 years, 8 months ago

In Mumbai region there is Pre-RMO, as the no. of students taking part was much more. While the other regions in the country have RMO directly (including other-than-Mumbai Maharashtra part. We are going to give it on 1st December. Let's discuss some good problems thereafter...!!!

Yash Jakhotiya - 7 years, 8 months ago

Promo paper

dhanesh shah - 3 years, 10 months ago

Priansh, can I get the copy of your PRMO paper??

CCD Pandhare - 7 years, 7 months ago

Well they didn't allow us to take it back home :/

Priyansh Sangule - 7 years, 6 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}`	$\boxed{123}$