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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.

Here is a list of problems (along with the names of the proposers) from Issue 1, 2008 of Mathematical Reflections that I like. The problem section actually contains 24 problems but I will post just a few here. $(1)$ The zeroes of the polynomial $P(x) = x^3 + x^2 + ax + b$ are all real and negative. Prove that $4a-9b \le 1$.

(Titu Andreescu) $(2)$ Let $x, y$ and $z$ be complex numbers such that $(y+z)(x-y)(x-z) = (z+x)(y-z)(y-x) = (x+y)(z-x)(z-y) = 1$ $= 1$.

Determine all possible values of $(x+y)(y+z)(z+x)$.

(Alex Anderson) $(3)$ Prove that there is no polynomial $P \in \mathbb{R}[X]$ of degree $n \ge 1$ such that $P(x) \in \mathbb{Q}$ for all real $x$.

(Ivan Borsenco) $(4)$ Let $P$ be a complex polynomial of degree $n > 2$, and let $A$ and $B$ be two complex matrices such that $AB \ne BA$ and $P(AB) = P(BA)$. Prove that $P(AB) = cI_2$ for some complex number $c$.

(Titu Andreescu and Dorin Andrica) $(5)$ Let $n = \prod_{i=1}^{k}p_i$ where $p_1, p_2, \ldots p_k$ are distinct odd primes. Prove that there is a $A \in M_n(\mathbb{Z})$ with $A^m = I_n$ iff the symmetric group $S_{n+k}$ has an element of order $m$.

(Jean-Charles Mathieux)

2008, Issue 1 of Mathematical Reflections is out! Check out the problem section.

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