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.