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1983 All Soviet Union Mathematical Olympiad
366
ASU 366 All Soviet Union MO 1983 Caratheodory's theorem with vectors and areas
ASU 366 All Soviet Union MO 1983 Caratheodory's theorem with vectors and areas
Source:
July 28, 2019
vector
geometry
areas
Problem Statement
Given a point
O
O
O
inside triangle
A
B
C
ABC
A
BC
. Prove that
S
A
∗
O
A
→
+
S
B
∗
O
B
→
+
S
C
∗
O
C
→
=
0
→
S_A * \overrightarrow{OA} + S_B * \overrightarrow{OB} + S_C * \overrightarrow{OC} = \overrightarrow{0}
S
A
∗
O
A
+
S
B
∗
OB
+
S
C
∗
OC
=
0
where
S
A
,
S
B
,
S
C
S_A, S_B, S_C
S
A
,
S
B
,
S
C
denote areas of triangles
B
O
C
,
C
O
A
,
A
O
B
BOC, COA, AOB
BOC
,
CO
A
,
A
OB
respectively.
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