The Hilbert Problems are a set of 23 mathematical challenges presented by the German mathematician David Hilbert in 1900. Aimed at defining future research directions, these problems span various areas of mathematics. While some were clearly stated, others were more open-ended. Over the decades, a majority of these problems have been resolved, significantly advancing mathematical knowledge. Some led to the establishment of entire new fields of study. Unresolved - as of AE53 - Hilbert problems:

Mathematical Treatment of the Axioms of Physics

Rigorous Foundation of Schubert's Enumerative Calculus

Topology of Algebraic Curves and Surfaces

Analysis of Solutions of Variational Problems

The Millennium Problems are seven mathematical challenges established by the Clay Mathematics Institute in 2000, each with a prize of one million dollars for a correct solution. These problems represent some of the most persistent and critical unsolved issues in mathematics, spanning areas such as number theory, geometry, analysis, and fluid dynamics. Only one of the problems, the Poincaré Conjecture, has been solved so far. Unresolved - as of AE53 - ones are:

Birch and Swinnerton-Dyer Conjecture

Hodge Conjecture

Navier–Stokes Existence and Smoothness

P vs NP Problem

Riemann Hypothesis

Yang–Mills Existence and Mass Gap

Divide in couples and discuss with Your colleagues cases when You use the word problem in daily life. Pose Yourself questions like "What was the last problem I solved ?", "What was the last problem I didn't manage to solve "? etc.

From Middle English probleme, from Middle French probleme, from Latin problema, from Ancient Greek πρόβλημα (próblēma, “anything thrown forward, hindrance, obstacle, anything projecting, a headland, promontory”), from προβάλλω (probállō, “to throw or lay something in front of someone, to put forward”), from προ- (pro-, “in front of”) + βάλλω (bállō, “to throw, to cast, to hurl”).

[note the etymologic similarity to word "project":  from Latin projectum ‘something prominent’, neuter past participle of proicere ‘throw forth’, from pro- ‘forth’ + jacere ‘to throw’)]