Let $K$ be a number field with ring of integers $\mathcal O_K$, let $\mathfrak m$ be an ideal in $\mathcal O_K$ and let $a \in \mathcal O_K$ such that $(a, \mathfrak m) = 1$.
Does there necessarily exist an $x \in a + \mathfrak m$ such that $x$ is totally positive? That is, for every real embedding $\sigma: \mathcal O_K \to \mathbb R$, $\sigma(x) > 0$.
Note that squares are always totally positive and therefore this is true whenever $a \in (\mathcal O_k/\mathfrak m)^2.$ This lets us assume that $(2, \mathfrak m) = 1$ by the Chinese remainder theorem.