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Fixed point exists except when...?

Source: Own. Malaysian APMO CST 2024 P2

February 24, 2024

geometry

Problem Statement

Let

k>1

. Fix a circle

\omega

with center

O

and radius

r

, and fix a point

A

with

\omega

.
Let

AB

,

AC

be tangents to

\omega

. Choose a variable point

P

on the minor arc

BC

in

\omega

. Lines

AB

and

CP

intersect at

Z

and lines

AC

and

k>1

intersect at

Y

. The circles

(BPX)

and

(CPY)

meet at another point

Z

.
Prove that the line

PZ

always passes through a fixed point except for one value of

k>1

, and determine this value.
Proposed by Ivan Chan Kai Chin

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