There's a convex 3n-polygon on the plane with a robot on each of it's vertices. Each robot fires a laser beam toward another robot. On each of your move,you select a robot to rotate counter clockwise until it's laser point a new robot. Three robots A, B and C form a triangle if A's laser points at B, B's laser points at C, and C's laser points at A. Find the minimum number of moves that can guarantee n triangles on the plane.

(2016 Taiwan TST round 3 problem 2-2)

Find all polynomials P(x) with integer coefficients such that for all real numbers s and t, if P(s) and P(t) are both integers, then P(st) is also an integer.

(2018 APMO problem 5)

Consider a 2018*2019 board with integers in each unit square. Two unit squares are said to be neighbors if they share a common edge. In each turn, you choose some unit squares. Then for each chosen unit square the average of all its neighbors is calculated. Finally, after these calculations are done, the number in each chosen unit square is replaced by the corresponding average.

Is it always possible to make the numbers in all squares become the same after finitely many turns?

(2019 APMO problem 4)

Chapter 4 is done by me! In this chapter, I developed an interesting technique using calculus/analysis/differential equation methods to solve functional equations. The main idea is to first prove monotonicity (so most of the problems in this chapter are R+ FEs) and use Lebesgue monotone differential theorem. There are even some problems that has no elementary solution yet!

Wanted: First one sending me an elementary solution of Example 50 (without using differentiability) will win $20 USD :)

Download   AoPS   Chapter 4 (Chinese)

Download (Chinese)

Download (Chinese)