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National and Regional Contests
China Contests
XES Mathematics Olympiad
the 10th XMO
the 10th XMO
Part of
XES Mathematics Olympiad
Subcontests
(1)
2
1
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PK bisects BC if AP _|_ OH
Given acute triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with orthocenter
H
H
H
and circumcenter
O
O
O
(
O
≠
H
O \ne H
O
=
H
) . Let
Γ
\Gamma
Γ
be the circumcircle of
△
B
O
C
\vartriangle BOC
△
BOC
. Segment
O
H
OH
O
H
untersects
Γ
\Gamma
Γ
at point
P
P
P
. Extension of
A
O
AO
A
O
intersects
Γ
\Gamma
Γ
at point
K
K
K
. If
A
P
⊥
O
H
AP \perp OH
A
P
⊥
O
H
, prove that
P
K
PK
P
K
bisects
B
C
BC
BC
. https://cdn.artofproblemsolving.com/attachments/a/b/267053569c41692f47d8f4faf2a31ebb4f4efd.png