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All-Russian Olympiad Regional Round
1998 All-Russian Olympiad Regional Round
10.6
10.6
Part of
1998 All-Russian Olympiad Regional Round
Problems
(1)
is pentagon convex? - All-Russian MO 1998 Regional (R4) 10.6
Source:
9/17/2024
The pentagon
A
1
A
2
A
3
A
4
A
5
A_1A_2A_3A_4A_5
A
1
A
2
A
3
A
4
A
5
contains bisectors
ℓ
1
\ell_1
ℓ
1
,
ℓ
2
\ell_2
ℓ
2
,
.
.
.
...
...
,
ℓ
5
\ell_5
ℓ
5
of angles
∠
A
1
\angle A_1
∠
A
1
,
∠
A
2
\angle A_2
∠
A
2
,
.
.
.
...
...
,
∠
A
5
\angle A_5
∠
A
5
respectively. Bisectors
ℓ
1
\ell_1
ℓ
1
and
ℓ
2
\ell_2
ℓ
2
intersect at point
B
1
B_1
B
1
,
ℓ
2
\ell_2
ℓ
2
and
ℓ
3
\ell_3
ℓ
3
- at point
B
2
B_2
B
2
, etc.,
ℓ
5
\ell_5
ℓ
5
and
ℓ
1
\ell_1
ℓ
1
intersect at point
B
5
B_5
B
5
. Can the pentagon
B
1
B
2
B
3
B
4
B
5
B_1B_2B_3B_4B_5
B
1
B
2
B
3
B
4
B
5
be convex?
geometry
pentagon
convex