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Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine National Mathematical Olympiad
2023 Ukraine National Mathematical Olympiad
11.6
11.6
Part of
2023 Ukraine National Mathematical Olympiad
Problems
(1)
Weird angle conditions geo!
Source: Ukraine MO 2023 11.6
4/5/2023
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
geometry