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National and Regional Contests
China Contests
XES Mathematics Olympiad
the 9th XMO
2
2
Part of
the 9th XMO
Problems
(1)
9th XMO 2022 P2: Multi-center and a weird computation
Source: 9th XMO 2022
6/18/2022
Given a
△
A
B
C
\triangle ABC
△
A
BC
with circumcenter
O
O
O
and orthocenter
H
(
O
≠
H
)
H(O\ne H)
H
(
O
=
H
)
. Denote the midpoints of
B
C
,
A
C
BC, AC
BC
,
A
C
as
D
,
E
D, E
D
,
E
and let
D
′
,
E
′
D', E'
D
′
,
E
′
be the reflections of
D
,
E
D, E
D
,
E
w.r.t. point
H
H
H
, respectively. If lines
A
D
′
AD'
A
D
′
and
B
E
′
BE'
B
E
′
meet at
K
K
K
, compute
K
O
K
H
\frac{KO}{KH}
KH
K
O
.
geometry
computation
China