Let $A$ be an $n\times n$ symmetric matrix with entries in $\mathbf F_2$ (the field with 2 elements, also referred to as $\mathbb Z/2\mathbb Z$). Prove that the diagonal of $A$ is in the span of its columns.
If the diagonal is only zeros the problem is trivial, but I haven't made any significant progress in the general case...
Consider the natural bilinear form $\cdot$ on $\mathbb{F}_2^n$.
Note that for all $v,w \in \mathbb{F}_2^n$, $(Av) \cdot w = (Aw) \cdot v$.
As a consequence, if $R$ is the range of $A$ and $K$ is the kernel of $A$, $R^{\perp} \subset K$. Since $\dim\,R^{\perp}=n-\dim\,R=\dim\,K$, $R^{\perp}=K$.
Note that this implies that $K^{\perp}=(R^{\perp})^{\perp} = R$ (for the second equality, the dimension equality is easy and one inclusion also is).
So what you need to show is that the diagonal (denoted $d$) of $A$ is orthogonal to any vector of $K$.
Now, if $Ax=0$, then $x^TAx=0$, and, since we work in $\mathbb{F}_2^n$, $x^TAx$ is $d \cdot x$.