Find all positive solutions of the system of equations
$x_1+x_2=(x_3)^2$ , $x_2+x_3=(x_4)^2$ , $x_3+x_4=(x_5)^2$ , $x_4+x_5=(x_1)^2$ , $x_5+x_1=(x_2)^2$
What i have done :
$(x_1+x_2)+(x_2+x_3)+(x_3+x_4)+(x_4+x_5)+(x_5+x_1)=(x_3)^2+(x_4)^2+(x_5)+(x_1)^2+(x_2)^2$
Thus $ 2(x_1+x_2+x_3+x_4+x_5)=(x_1)^2+(x_2)^2+(x_3)^2+(x_4)^2+(x_5)^2$
Let $ x = (x_1,x_2,x_3,x_4,x_5) $ . I am currently attempting to find $ x$ such that $||x||$ is at maximum.By symmetry $||x||$ attains maximum when $ x_1 = x_2 =x_3 =x_4 = x_ 5 $. Thus $ 2(5x_1) =5(x_1)^2 $. Hence $x_1=x_2 =x_3=x_4=x_5=2 $.I am unsure how to tackle this problem from this points onward.
Any insight is greatly appreciated !
Thanks !