Yeah, they do often require a lot of math. It's just a restricted set of topics. The IMO doesn't do any calculus or anything beyond calculus like real analysis, topology, etc. It does have a lot of geometry problems that use geometry that typically isn't in the American curriculum, and it has topics like number theory that aren't really in a standard high school curriculum either. But the topics are restricted so that they are pretty close to understandable by someone who has just done well in a standard high school mathematics curriculum.

Here's an example. For the problem they include in the article:

Let n be an integer greater than or equal to 3. Prove that there is a set of n points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.

This is a whole lot easier if you're familiar with Pick's Theorem, which gives you a formula for the area of a polygon whose corners are integer lattice points. In particular, any three integer lattice points define a triangle with rational area. So all you have to do is find n points that aren't colinear with each other and have irrational distance, and you'll solve the problem.

Technically, you could do this without Pick's Theorem, because you'll be able to find an equivalent formula for the area of these triangles. But it's much easier if you do know Pick's Theorem, and that's the sort of thing that you typically don't pick up in standard high school math, but you would pick up if you were practicing a bunch of different Olympiad-type problems.