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International Contests
Kvant Problems
Kvant 2023
M2756
M2756
Part of
Kvant 2023
Problems
(1)
Determine if there exists a convex hexagon $ABCDEF$ with
Source: Caucasus MO 2023
7/16/2023
a) Determine if there exists a convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
with
∠
A
B
D
+
∠
A
E
D
>
18
0
∘
,
\angle ABD + \angle AED > 180^{\circ},
∠
A
B
D
+
∠
A
E
D
>
18
0
∘
,
∠
B
C
E
+
∠
B
F
E
>
18
0
∘
,
\angle BCE + \angle BFE > 180^{\circ},
∠
BCE
+
∠
BFE
>
18
0
∘
,
∠
C
D
F
+
∠
C
A
F
>
18
0
∘
.
\angle CDF + \angle CAF > 180^{\circ}.
∠
C
D
F
+
∠
C
A
F
>
18
0
∘
.
b) The same question, with additional condition, that diagonals
A
D
,
B
E
,
AD, BE,
A
D
,
BE
,
and
C
F
CF
CF
are concurrent.
geometry