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Contests
National and Regional Contests
China Contests
China Team Selection Test
2003 China Team Selection Test
2
Find a sequence
Find a sequence
Source: China TST 2003
June 29, 2006
algebra unsolved
algebra
Problem Statement
Given an integer
a
1
a_1
a
1
(
a
1
≠
−
1
a_1 \neq -1
a
1
=
−
1
), find a real number sequence
{
a
n
}
\{ a_n \}
{
a
n
}
(
a
i
≠
0
,
i
=
1
,
2
,
⋯
,
5
a_i \neq 0, i=1,2,\cdots,5
a
i
=
0
,
i
=
1
,
2
,
⋯
,
5
) such that
x
1
,
x
2
,
⋯
,
x
5
x_1,x_2,\cdots,x_5
x
1
,
x
2
,
⋯
,
x
5
and
y
1
,
y
2
,
⋯
,
y
5
y_1,y_2,\cdots,y_5
y
1
,
y
2
,
⋯
,
y
5
satisfy
b
i
1
x
1
+
b
i
2
x
2
+
⋯
+
b
i
5
x
5
=
2
y
i
b_{i1}x_1+b_{i2}x_2+\cdots +b_{i5}x_5=2y_i
b
i
1
x
1
+
b
i
2
x
2
+
⋯
+
b
i
5
x
5
=
2
y
i
,
i
=
1
,
2
,
3
,
4
,
5
i=1,2,3,4,5
i
=
1
,
2
,
3
,
4
,
5
, then
x
1
y
1
+
x
2
y
2
+
⋯
+
x
5
y
5
=
0
x_1y_1+x_2y_2+\cdots+x_5y_5=0
x
1
y
1
+
x
2
y
2
+
⋯
+
x
5
y
5
=
0
, where
b
i
j
=
∏
1
≤
k
≤
i
(
1
+
j
a
k
)
b_{ij}=\prod_{1 \leq k \leq i} (1+ja_k)
b
ij
=
∏
1
≤
k
≤
i
(
1
+
j
a
k
)
.
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