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IberoAmerican
2024 Iberoamerican
2
2
Part of
2024 Iberoamerican
Problems
(1)
American point shows in Ibero geo!
Source: Iberoamerican MO 2024 Day 1 P2
9/21/2024
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute triangle and let
M
,
N
M, N
M
,
N
be the midpoints of
A
B
,
A
C
AB, AC
A
B
,
A
C
respectively. Given a point
D
D
D
in the interior of segment
B
C
BC
BC
with
D
B
<
D
C
DB<DC
D
B
<
D
C
, let
P
,
Q
P, Q
P
,
Q
the intersections of
D
M
,
D
N
DM, DN
D
M
,
D
N
with
A
C
,
A
B
AC, AB
A
C
,
A
B
respectively. Let
R
≠
A
R \ne A
R
=
A
be the intersection of circumcircles of triangles
△
P
A
Q
\triangle PAQ
△
P
A
Q
and
△
A
M
N
\triangle AMN
△
A
MN
. If
K
K
K
is midpoint of
A
R
AR
A
R
, prove that
∠
M
K
N
=
2
∠
B
A
C
\angle MKN=2\angle BAC
∠
M
K
N
=
2∠
B
A
C
geometry
circumcircle