Solve this system (Canadian Math Olympiad,1996)?

Question

Find all triples (x,y,z) of real numbers x,y,z satisfying the following system of equations:

(4*x^2)/(1+4*x^2)=y,

(4*y^2)/(1+4*y^2)=z,

(4*z^2)/(1+4*z^2)=x.

A "*" stands for multiplication.

Only justified answers are welcome. Thank you.

Answer 1

x=y=z=0 seems to be the only solution.

Each x,y and z is +ve and <1. and clearly they are equal.

Answer 2

let  f(x)=4x^2/(1+4x^2)

you can prove  that   f(f(f(x)))=x

then we have only  solution   when   f(x)=x

4x^2/(1+4x^2)=x     x(1+4x^2)=4x^2      x+4x^3-4x^2=0

x(4x^2-4x+1)=0   then  x=0   4x^2-4x+1=0      (2x-1)^2=0  2x-1=0

2x=1  x=1/2

x=0  then  y=0    z=0

x=1/2   y=1/2   z=1/2

you have   (0,0,0)  or  (1/2,1/2,1/2)