\(Q1\) Find the largest y such that

\(\frac { 1 }{ 1+{ x }^{ 2 } } \ge \frac { y-x }{ y+x } \quad for\quad all\quad x>0\)

\(Q2\) Find the minimum and maximum values of

\(\frac { x+1 }{ xy+x+1 } +\frac { y+1 }{ yz+y+1 } +\frac { z+1 }{ zx+z+1 } \)

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## Comments

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TopNewestApplying componendo and dividendo we get \[\frac{2+x^{2}}{-x^{2}}\geq\frac{2y}{-2x}\]

\[\frac{2}{x}+x \geq y \]

Applying \(A.M-G.M\)

\[\frac{2}{x}+x\geq 2\sqrt{2}\]

Therefore \(y=2\sqrt{2}\).

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Thanks. How could I forget to tag you! Btw can you even have a look at Olympiad corner#1 q1

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Answered

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@Dev Sharma @Adarsh Kumar @Surya Prakash @Svatejas Shivakumar @Kushagra Sahni

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@Satyajit Ghosh Thank you for mentioning me!

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Therefore, the maximum value of \(y\) is \(\sqrt {8}\)

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Thanks! Can you tell the answer for q2. Do check out my Olympiad corner#1 which has q1 unanswered.

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Do u think the 2nd question is correct? Check the expression again from your book.

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If you have got the answer, please give a hint at least

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Sorry the denominator of 1st term had xy+y+1 but now I have updated it it to the correct question

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Lol both questions are Q1?

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I've fixed it.

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