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National and Regional Contests
China Contests
China Team Selection Test
2010 China Team Selection Test
2
China TST 2010, Problem 2
China TST 2010, Problem 2
Source:
August 28, 2010
algebra
polynomial
function
complex numbers
algebra unsolved
Problem Statement
Let
A
=
{
a
1
,
a
2
,
⋯
,
a
2010
}
A=\{a_1,a_2,\cdots,a_{2010}\}
A
=
{
a
1
,
a
2
,
⋯
,
a
2010
}
and
B
=
{
b
1
,
b
2
,
⋯
,
b
2010
}
B=\{b_1,b_2,\cdots,b_{2010}\}
B
=
{
b
1
,
b
2
,
⋯
,
b
2010
}
be two sets of complex numbers. Suppose
∑
1
≤
i
<
j
≤
2010
(
a
i
+
a
j
)
k
=
∑
1
≤
i
<
j
≤
2010
(
b
i
+
b
j
)
k
\sum_{1\leq i<j\leq 2010} (a_i+a_j)^k=\sum_{1\leq i<j\leq 2010}(b_i+b_j)^k
1
≤
i
<
j
≤
2010
∑
(
a
i
+
a
j
)
k
=
1
≤
i
<
j
≤
2010
∑
(
b
i
+
b
j
)
k
holds for every
k
=
1
,
2
,
⋯
,
2010
k=1,2,\cdots, 2010
k
=
1
,
2
,
⋯
,
2010
. Prove that
A
=
B
A=B
A
=
B
.
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