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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 .
Let and be complex numbers such that
Determine all possible values of .
Prove that there is no polynomial of degree such that for all real .
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 .