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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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July 2021