Here is a challenging problem from the Polish Mathematical Olympiads published in 1960.

“22. Prove that the polynomial

x⁴⁴ + x³³ + x²² + x¹¹ + 1

is divisible by the polynomial

x⁴ + x³ + x² + x + 1.”

See the Polynomial Division Problem

(Update 8/23/2021)  The idea expressed in this post that mathematicians are “lazy” and seek short-cuts to solving questions and problems, as I did in this one, was recently the subject of a Numberphile post by Marcus du Sautoy: “Mathematics is all about SHORTCUTS“.