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$\large \dfrac{4x^2}{1+4x^2}=y,\qquad \dfrac{4y^2}{1+4y^2}=z,\qquad \dfrac{4z^2}{1+4z^2}=x$

Let $x,y,z$ be non-zero real number satisfying the system of equations above. Find the number of triplets $(x,y,z)$ satisfying these conditions.

#Algebra

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`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`	$\sum_{i=1}^3$
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Comments

Use AM-GM. $y=\frac{4x^2}{4x^2+1}=<\frac{4x^2}{2\sqrt{4x^2}}=x$ Therefore $x>=y$ similarily we get for other ones ending with: $x>=y>=z>=x$ which leaves us with $x=y=z$ Obivious solution is $x=0$ and other one is $x=\frac{1}{2}$

Dragan Marković - 5 years ago

Have you checked AM,GM and HM of the three expressions?

Akshay Yadav - 5 years ago

1+2=7

Chua Hsuan - 5 years 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}`	$\boxed{123}$