Let $Q(x):\mathbb{R}^4 \to \mathbb{R}^4$ - Surjective function.
$$Q(x) = \begin{pmatrix} q_1(x_1,x_2) \\ q_2(x_2,x_3) \\ q_3(x_3,x_4) \\ q_4(x_4,x_1) \end{pmatrix}$$
where $q_i(x)$ is a quadratic form.
Does it exist $x \neq 0$ such that $Q(x)=0 $ ?
I tried to solve it by system of linear equations, using surjective property..without results... So, maybe have any ideas, answers?