If $A\in\mathbb{R}^{n\times n}, A>0$ (symmetric, positive definite), prove that there exist $x=\begin{bmatrix} \sigma_1 \\ \vdots \\ \sigma_n\end{bmatrix}$, $\sigma_i\in\{\pm1\}$, such that $(Ax)_i\neq0$ for all $i$.
Example: If $A=\begin{bmatrix}2 & -1 & -1\\-1 & 2 &1 \\-1 & 1 &2\end{bmatrix}>0$, then $Ax=\begin{bmatrix}0\\2\\2\end{bmatrix}$ for $x=\begin{bmatrix}1\\1\\1\end{bmatrix}$,
but $Ax=\begin{bmatrix}4\\-4\\-4\end{bmatrix}$ for $x=\begin{bmatrix}1\\-1\\-1\end{bmatrix}$.
From simulation of $3\times3$ and $4\times4$ matrices, it turns out to be true.