BALTIC-WAY 1992 Problem 10

Find all fourth degree polynomials $p(x)$ such that the following four conditions are satisfied:

(i) $p(x)=p(-x)$ for all $x$.

(ii) $p(x) \geq 0$ for all $x$.

(iii) $p(0)=1$.

(iv) $p(x)$ has exactly two local minimum points $x_{1}$ and $x_{2}$ such that $\left|x_{1}-x_{2}\right|=2$.