1

Part of 2023 China Second Round

Problems(2)

A problem of Complex numbers

Source: 2023 China Second Round Test 1,P1

We define a complex number

z=9+10i

please find the maximum of a positive integer

n

which satisfies

|z^n|\leq2023

complex numbersalgebra

Fixed Circumcenter

Source: Problem 1, China National High School Mathematics League 2023

A,B

be two fixed points on a plane and

\Omega

a fixed semicircle arc with diameter

AB

T

be another fixed point on

\Omega

\omega

a fixed circle that passes through

A

T

and has its center in

\Delta ABT

P

be a moving point on the arc

TB

(endpoints excluded), and

C,D

be two moving points on

\omega

C

lies on segment

AP

C,D

lies on different sides of line

AB

CD\ \bot \ AB

. Denote the circumcenter of

\Delta CDP

K

. Prove that (i)

K

lies on the circumcircle of

\Delta TDP

K

is a fixed point.

geometrycircumcirclechang09