# Help!

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

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.

Note by Akshat Sharda
1 year, 11 months ago

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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 $$x=\frac{1}{2}$$

- 1 year, 11 months ago

1+2=7

- 1 year, 10 months ago

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

- 1 year, 11 months ago