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

Problem: A polynomial is called a “mirror” if . Let and consider polynomials such that , and . Prove that is a mirror iff 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.

The zeroes of the polynomial are all real and negative. Prove that .

(*Titu Andreescu*)

Let and be complex numbers such that

.

Determine all possible values of .

(*Alex Anderson*)

Prove that there is no polynomial of degree such that for all real .

(*Ivan Borsenco*)

Let be a complex polynomial of degree , and let and be two complex matrices such that and . Prove that for some complex number .

(*Titu Andreescu* and *Dorin Andrica*)

Let where are distinct odd primes. Prove that there is a with iff the symmetric group has an element of order .

(*Jean-Charles Mathieux*)

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

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