1. Combinatorial mathematics problems: These problems involve complex counting and permutation and combination, and need to deeply understand the basic principles of probability theory and statistics. For example, given n different elements, how many different combinations of * * *?

2. Number theory problems: These problems mainly involve the properties and theorems of integers, such as prime numbers, congruences, Fermat's last theorem, etc. For example, the following example solves Goldbach's conjecture.

3. Graph theory problems: These problems involve the structure and properties of graphs, such as Euler paths and Hamiltonian loops. For example, in an undirected graph, starting from vertex A, passing through all other vertices once, and then returning to A, how many paths are there?

4. Geometric problems: This kind of problems involves the calculation of the shape, area and volume of space geometry, such as polyhedron and rotator in solid geometry. For example, the surface area of a cube is 6a^2, where a is a positive integer. Find the volume of a cube.

5. Dynamic programming problem: This kind of problem involves state transition and the search for the optimal solution, and it is usually necessary to establish equations through recursive relations. For example, if a person wants to start from the origin of coordinates, he can move one grid to the right or upward at a time, and find the path with the largest sum of path lengths among all paths with coordinates (m, n).

6. Calculus problems: These problems involve the concepts and calculation methods of limit, derivative and integral of functions. For example, solve the following definite integral problem.

7. Linear algebraic problems: These problems involve concepts and operations such as vectors, matrices and linear equations. For example, given that matrix A satisfies a 2-2a+I = 0, find the maximum eigenvalue of A and the corresponding eigenvector.

These problems need profound mathematical knowledge, strong logical thinking ability and rich problem-solving skills, which have played a very good role in cultivating students' interest in mathematics and improving their problem-solving ability.