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)