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All-Russian Olympiad
1986 All Soviet Union Mathematical Olympiad
434
434
Part of
1986 All Soviet Union Mathematical Olympiad
Problems
(1)
ASU 434 All Soviet Union MO 1986 vector sum zero in n-gon A_1A_2...A_n
Source:
8/7/2019
Given a regular
n
n
n
-gon
A
1
A
2
.
.
.
A
n
A_1A_2...A_n
A
1
A
2
...
A
n
. Prove that if a)
n
n
n
is even number, than for the arbitrary point
M
M
M
in the plane, it is possible to choose signs in an expression
±
M
A
1
→
±
M
A
2
→
±
.
.
.
±
M
A
n
→
\pm \overrightarrow{MA_1} \pm \overrightarrow{MA_2} \pm ... \pm \overrightarrow{MA_n}
±
M
A
1
±
M
A
2
±
...
±
M
A
n
to make it equal to the zero vector . b)
n
n
n
is odd, than the abovementioned expression equals to the zero vector for the finite set of
M
M
M
points only.
vector
polygon
geometry