I need to solve the equation system
$x_1(x_1-1)=3x_2-2^2$
$x_2(x_2-1)=5x_3-3^2$
$x_3(x_3-1)=7x_4-4^2$
...
$x_9(x_9-1)=19x_{10}-10^2$
$x_{10}(x_{10}-1)=x_1-1^2$
I have generalized it to
$x_n(x_n-1)=(2n+1)x_{n+1}-(n+1)^2$
until I get to $x_{10}$. But then I don't know what to do... EDIT: typo
HINT:
Assume $x_i\in \mathbb{R},\;i=1,\dots,10.\;$ Adding the $10$ equations and putting all terms to the left side gives $$(x_1-1)^2+(x_2-2)^2+\cdots (x_{10}-10)^2=0,$$
which is only possible if ... (you can finish, I think).