Informal "real-world" mathematical problems are questions related to a concrete setting, such as "Adam has five apples and gives John three.
While mathematicians usually study them for their own sake, by doing so, results may be obtained that find application outside the realm of mathematics.
Some abstract problems have been rigorously proved to be unsolvable, such as squaring the circle and trisecting the angle using only the compass and straightedge constructions of classical geometry, and solving the general quintic equation algebraically.
Mathematics educators using problem solving for evaluation have an issue phrased by Alan H. Schoenfeld: The same issue was faced by Sylvestre Lacroix almost two centuries earlier: Such degradation of problems into exercises is characteristic of mathematics in history.
For example, describing the preparations for the Cambridge Mathematical Tripos in the 19th century, Andrew Warwick wrote: