PUMaC 2025 · 代数（A 组） · 第 7 题

7. Let

π π iπ/ 2025 ω = e = cos + i sin . 2025 2025 2 4049 David writes the complex numbers 2 , ω, ω , . . . , ω in a line going from left to right on a blackboard. Then, each minute, if the two leftmost complex numbers on the blackboard are z and z , David erases the two numbers and replaces them with 1 2 1 1 − . 1 1 z + z + 1 2 z z 1 2 After 4049 minutes, there is only one number remaining. What is the last remaining number?