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Putnam
1996 Putnam
6
Putnam 1996 B6
Putnam 1996 B6
Source:
June 6, 2014
Putnam
college contests
Problem Statement
Let
(
a
1
,
b
1
)
,
(
a
2
,
b
2
)
,
…
,
(
a
n
,
b
n
)
(a_1,b_1),(a_2,b_2),\ldots ,(a_n,b_n)
(
a
1
,
b
1
)
,
(
a
2
,
b
2
)
,
…
,
(
a
n
,
b
n
)
be the vertices of a convex polygon containing the origin in its interior. Prove that there are positive real numbers
x
,
y
x,y
x
,
y
such that :
(
a
1
,
b
1
)
x
a
1
y
b
1
+
(
a
2
,
b
2
)
x
a
2
y
b
2
+
…
+
(
a
n
,
b
n
)
x
a
n
y
b
n
=
(
0
,
0
)
(a_1,b_1)x^{a_1}y^{b_1}+(a_2,b_2)x^{a_2}y^{b_2}+\ldots +(a_n,b_n)x^{a_n}y^{b_n}=(0,0)
(
a
1
,
b
1
)
x
a
1
y
b
1
+
(
a
2
,
b
2
)
x
a
2
y
b
2
+
…
+
(
a
n
,
b
n
)
x
a
n
y
b
n
=
(
0
,
0
)
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