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Taiwan Mathematics Olympiad
2023 Taiwan Mathematics Olympiad
3
3
Part of
2023 Taiwan Mathematics Olympiad
Problems
(1)
A weird conditional geometry
Source: 2023 Taiwan Mathematics Olympiad
2/8/2023
Let
O
O
O
be the center of circle
Γ
\Gamma
Γ
, and
A
A
A
,
B
B
B
be two points on
Γ
\Gamma
Γ
so that
O
,
A
O, A
O
,
A
and
B
B
B
are not collinear. Let
M
M
M
be the midpoint of
A
B
AB
A
B
. Let
P
P
P
and
Q
Q
Q
be points on
O
A
OA
O
A
and
O
B
OB
OB
, respectively, so that
P
≠
A
P \neq A
P
=
A
and
P
,
M
,
Q
P, M, Q
P
,
M
,
Q
are collinear. Let
X
X
X
be the intersection of the line passing through
P
P
P
and parallel to
A
B
AB
A
B
and the line passing through
Q
Q
Q
and parallel to
O
M
OM
OM
. Let
Y
Y
Y
be the intersection of the line passing through
X
X
X
and parallel to
O
A
OA
O
A
and the line passing through
B
B
B
and orthogonal to
O
X
OX
OX
. Prove that: if
X
X
X
is on
Γ
\Gamma
Γ
, then
Y
Y
Y
is also on
Γ
\Gamma
Γ
.Proposed by usjl
Taiwan
geometry