Let $\{F_i\}$ be given quadratic polynomials in $\mathbb Q[x_1,\ldots,x_n]$. If for every linear polynomial $l$, there exists linear polynomials $\{l_i\}$ (depend on $l$) such that
(i) $\sum l_i F_i$ is devisible by $l$.
(ii) Not every $l_i$ is a multiple of $l$.
Is it true that $\sum l_i F_i=0$ for some $\{l_i\}$?
I have no idea how to approach this. Any comments would be helpful.