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Balkan MO Shortlist
2022 Balkan MO Shortlist
G5
G5
Part of
2022 Balkan MO Shortlist
Problems
(1)
Circle geo
Source: BMO Shortlist 2022, G5
5/13/2023
Let
A
B
C
ABC
A
BC
be a triangle with circumcircle
ω
\omega
ω
, circumcenter
O
O{}
O
, and orthocenter
H
H{}
H
. Let
K
K{}
K
be the midpoint of
A
H
AH{}
A
H
. The perpendicular to
O
K
OK{}
O
K
at
K
K{}
K
intersects
A
B
AB{}
A
B
and
A
C
AC{}
A
C
at
P
P{}
P
and
Q
Q{}
Q
, respectively. The lines
B
K
BK
B
K
and
C
K
CK
C
K
intersect
ω
\omega
ω
again at
X
X{}
X
and
Y
Y{}
Y
, respectively. Prove that the second intersection of the circumcircles of triangles
K
P
Y
KPY
K
P
Y
and
K
Q
X
KQX
K
QX
lies on
ω
\omega
ω
.Stefan Lozanovski
geometry