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Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine National Mathematical Olympiad
2023 Ukraine National Mathematical Olympiad
9.6
9.6
Part of
2023 Ukraine National Mathematical Olympiad
Problems
(1)
Angle condition again
Source: Ukraine MO 2023 9.6
4/5/2023
A point
O
O
O
lies inside
△
A
B
C
\triangle ABC
△
A
BC
so that
∠
B
O
C
=
90
−
∠
B
A
C
\angle BOC=90-\angle BAC
∠
BOC
=
90
−
∠
B
A
C
. Let
B
O
,
C
O
BO, CO
BO
,
CO
meet
A
C
,
A
B
AC, AB
A
C
,
A
B
at
K
,
L
K, L
K
,
L
. Points
K
1
,
L
1
K_1, L_1
K
1
,
L
1
lie on the segments
C
L
,
B
K
CL, BK
C
L
,
B
K
so that
K
1
B
=
K
1
K
K_1B=K_1K
K
1
B
=
K
1
K
and
L
1
C
=
L
1
L
L_1C=L_1L
L
1
C
=
L
1
L
. If
M
M
M
is the midpoint of
B
C
BC
BC
, then prove that
∠
K
1
M
L
1
=
9
0
o
\angle K_1ML_1=90^{o}
∠
K
1
M
L
1
=
9
0
o
.Proposed by Anton Trygub
geometry