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2023 Ukraine National Mathematical Olympiad
11.6
Weird angle conditions geo!
Weird angle conditions geo!
Source: Ukraine MO 2023 11.6
April 5, 2023
geometry
Problem Statement
Let
K
K
K
be the midpoint of the median
A
M
AM
A
M
of a triangle
A
B
C
ABC
A
BC
. Points
X
,
Y
X, Y
X
,
Y
lie on
A
B
,
A
C
AB, AC
A
B
,
A
C
, respectively, such that
∠
K
X
M
=
∠
A
C
B
\angle KXM =\angle ACB
∠
K
XM
=
∠
A
CB
,
A
X
>
B
X
AX>BX
A
X
>
BX
and similarly
∠
K
Y
M
=
∠
A
B
C
\angle KYM =\angle ABC
∠
K
Y
M
=
∠
A
BC
,
A
Y
>
C
Y
AY>CY
A
Y
>
C
Y
. Prove that
B
,
C
,
X
,
Y
B, C, X, Y
B
,
C
,
X
,
Y
are concyclic.Proposed by Mykhailo Shtandenko
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