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Annoying algebra in today's pre-RMO !

Hello !

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

If

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

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 .

Note by Priyansh Sangule
3 years, 10 months ago

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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 · 3 years, 10 months ago

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@Mark Hennings Thank You ! Priyansh Sangule · 3 years, 10 months ago

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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 · 3 years, 10 months ago

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@Advitiya Brijesh i thought same . quite easy. Umang Garg · 3 years, 9 months ago

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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 · 3 years, 10 months ago

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Priansh, can I get the copy of your PRMO paper?? Ccd Pandhare · 3 years, 10 months ago

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@Ccd Pandhare Well they didn't allow us to take it back home :/ Priyansh Sangule · 3 years, 9 months ago

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

Thanks! Pranav Arora · 3 years, 10 months ago

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@Pranav Arora Promo paper Dhanesh Shah · 1 week, 2 days ago

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@Pranav Arora 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 · 3 years, 10 months ago

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@Pranav Arora 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 · 3 years, 10 months ago

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@Priyansh Sangule 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 · 3 years, 10 months ago

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