Based on some experiments, I find that the following two statements are correct. But I can not prove this. At the same time, I still can not find the counterexmaples.

Let $p(x)=x^{n}+a_{2}x^{n-2}+a_{3}x^{n-3}+\dots+a_{n-1}x+a_{n}$ be a polynomial with interger coefficients, where $a_{k}\geq0$ for every even $k$ and $a_{k}\leq0$ otherwise. Note here that the coefficient of $x^{n-1}$ is equal to 0. Suppose that there exists some odd number $p$ such that $a_{p}<0$ and $a_{p-1}>0$. Then the following two statements should be true:

$p(x)$ has at most one nonzero real root.

$p(x)$ has no pure imaginary zeros, i.e. $p(x)$ has no zero in the form $\alpha\textrm{i}$, where $\alpha\neq0$ and $\textrm{i}^{2}=-1$.

Thanks for your time.

I am sorry for losing a condition that $a_{2}\geq5$.