Advanced mathematics is a challenging and interesting subject, which not only cultivates our logical thinking ability, but also makes us feel the charm of mathematics in the process of solving problems. The following are some interesting topics in advanced mathematics:

1. Prove that the diagonal length of a square is twice the root sign of its side length.

2. Let f(x) = x 3-3x 2+2x, and find the extreme point and inflection point of f(x).

3. Solve all the roots of the equation x 5-3x 4+2x 3-x 2+x-1= 0.

4. Let the function y=f(x) be continuous in the interval [a, b] and f(a)=f(b), and prove the existence of ξ∈(a, b) so that f'(ξ)=0.

5. Solve the following differential equation: dy/dx = (x 2-y 2)/(x 2+y 2).

6. Let the function f(x) be continuous in the interval [0, π] and f(0)=f(π), and prove the existence of c∈(0, π) so that f'(c)=0.

7. Solve the following integral problem: ∫ _ 0 π sin (x 2) dx.

8. Let f(x) be continuous in the interval [- 1, 1], and f(- 1)=f( 1), and prove that c∈(- 1,/kloc-).

9. Solve the following series convergence problem: ∑ _ {n =1} {infty} (1/n) n.

10. Let the function f(x) be continuous in the interval [0, 1], and f(0)=f( 1), which proves the existence of c∈(0, 1) and makes f' (c)