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All-Russian Olympiad
1990 All Soviet Union Mathematical Olympiad
526
526
Part of
1990 All Soviet Union Mathematical Olympiad
Problems
(1)
ASU 526 All Soviet Union MO 1990 n vectors with zero sum, interior points
Source:
8/14/2019
Given a point
X
X
X
and
n
n
n
vectors
x
i
→
\overrightarrow{x_i}
x
i
with sum zero in the plane. For each permutation of the vectors we form a set of
n
n
n
points, by starting at
X
X
X
and adding the vectors in order. For example, with the original ordering we get
X
1
X_1
X
1
such that
X
X
1
=
x
1
→
,
X
2
XX_1 = \overrightarrow{x_1}, X_2
X
X
1
=
x
1
,
X
2
such that
X
1
X
2
=
x
2
→
X_1X_2 = \overrightarrow{x_2}
X
1
X
2
=
x
2
and so on. Show that for some permutation we can find two points
Y
,
Z
Y, Z
Y
,
Z
with angle
∠
Y
X
Z
=
6
0
o
\angle YXZ = 60^o
∠
Y
XZ
=
6
0
o
, so that all the points lie inside or on the triangle
X
Y
Z
XYZ
X
Y
Z
.
vector
geometry
interior