If it's a math department, it's mainly analysis, algebra and geometry.
The high school algebra (including competitions) you mentioned is the foundation of advanced mathematics, and it should be used in all subjects. Algebra in college is completely different from algebra in high school. University algebra studies algebraic structure, sets and operations of their definitions, such as group ring fields. Specifically, you can look at linear algebra, advanced algebra, modern algebra and so on in universities.
Specializing in number theory will reach the postgraduate stage, and it is a major in basic mathematics. Number theory is divided into algebraic number theory and analytic number theory. China is strong in analytic number theory, and Chen Jingrun's 1+2 is analytic number theory, mainly studying calculus. Algebraic number theory mainly uses algebraic methods to study number theory, which is what I said earlier.
There is no plane geometry, because all the problems of plane geometry can be solved by analytic geometry, trigonometry, computer programming or other methods. If there are no remaining problems, I will not study them. But there are also geometry, mainly multidimensional analytic geometry (that is, linear algebra, because the abstraction of high-dimensional analytic geometry is algebra), topology and so on.
As for combinatorial mathematics, it should be said that it is a follow-up subject of algebra, and there are not many requirements in the undergraduate stage. If you want to study, it is also the content of graduate students. Graph theory is actually a combination of combinatorial mathematics and topology, and it is also the content of postgraduate entrance examination.
You may want to ask what to study in the undergraduate stage, but in fact you want to learn the necessary basic knowledge, such as calculus. The mathematics in the whole university has become a system, and you don't want to leave before high school, a little bit left, a little bit left. If you want to study what you say systematically, you must master the mathematics content of the undergraduate course. In addition, studying root competition is different. Research takes a lot of time to think, and the competition is only a few hours. So if you want to learn mathematics, you can't prepare almost all branches like a competition. You can only study some of them. After all, there are too many things in mathematics, so what you say is basically graduate students' things.