Binary solutions to quadratic form with indefinite matrix

Question

I have a $n\times n$ real symmetric indefinite non-zero matrix $A$ where diagonal elements are all the same.

Assume $x \in \{-1, 1\}^n$. There are two questions I'm tackling:

1. Is this statement true: there exists such an A where $x^TAx = 0, \forall x$.

2. Say I have found an $x$ such that $x^TAx = 0$. Is there an analytical (or computationally efficient) way to construct from $x$, a solution $\hat{x} \in \{-1,1\}^n$ such that $\hat{x}^T A \hat{x} \neq 0$?

There are related questions but the insights there didn't help me much: zeros of $x^*Ax$, a quadratic form Solution to a quadratic form

Any suggestions, will help. Thank you.