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 .