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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.

Note by Akshat Sharda
1 year, 6 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 \(x=\frac{1}{2}\)

Dragan Marković - 1 year, 6 months ago

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1+2=7

Chua Hsuan - 1 year, 5 months ago

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

Akshay Yadav - 1 year, 6 months ago

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