In an earlier issue of Mathematical Reflections, Iurie Boreico (from Harvard) proposed the following problem.

Problem: A polynomial $p \in \mathbb{R}[X]$ is called a “mirror” if $|p(x)| = |p(-x)|$ . Let $f \in \mathbb{R}[X]$ and consider polynomials $p, q \in \mathbb{R}[X]$ such that $p(x) - p'(x) = f(x)$, and $q(x) + q'(x) = f(x)$. Prove that $p + q$ is a mirror iff $f$ is a mirror.

Two solutions – the pdf file size is around 1 Mb – to the above problem were proposed, and the one by the author is very close to the one I had worked out myself (partially) earlier but never really got around finishing it. So, I will post my solution here but at a slightly later time. In the meantime, you might be interested in finding a solution yourself.