MathDB
Problems
Contests
National and Regional Contests
Malaysia Contests
Malaysian APMO Camp Selection Test
2024 Malaysian APMO Camp Selection Test
2
2
Part of
2024 Malaysian APMO Camp Selection Test
Problems
(1)
Fixed point exists except when...?
Source: Own. Malaysian APMO CST 2024 P2
2/24/2024
Let
k
>
1
k>1
k
>
1
. Fix a circle
ω
\omega
ω
with center
O
O
O
and radius
r
r
r
, and fix a point
A
A
A
with
O
A
=
k
r
OA=kr
O
A
=
k
r
. Let
A
B
AB
A
B
,
A
C
AC
A
C
be tangents to
ω
\omega
ω
. Choose a variable point
P
P
P
on the minor arc
B
C
BC
BC
in
ω
\omega
ω
. Lines
A
B
AB
A
B
and
C
P
CP
CP
intersect at
X
X
X
and lines
A
C
AC
A
C
and
B
P
BP
BP
intersect at
Y
Y
Y
. The circles
(
B
P
X
)
(BPX)
(
BPX
)
and
(
C
P
Y
)
(CPY)
(
CP
Y
)
meet at another point
Z
Z
Z
. Prove that the line
P
Z
PZ
PZ
always passes through a fixed point except for one value of
k
>
1
k>1
k
>
1
, and determine this value.Proposed by Ivan Chan Kai Chin
geometry