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Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2004 China Team Selection Test
1
System of Equations
System of Equations
Source: China TST 2004 Quiz
February 1, 2009
algebra
system of equations
algebra unsolved
Problem Statement
Given integer
n
n
n
larger than
5
5
5
, solve the system of equations (assuming
x
i
≥
0
x_i \geq 0
x
i
≥
0
, for
i
=
1
,
2
,
…
n
i=1,2, \dots n
i
=
1
,
2
,
…
n
):
{
x
1
+
2
2
x
2
+
3
2
x
3
+
⋯
+
n
2
x
n
=
n
+
2
,
x
1
+
2
2
x
2
+
3
2
x
3
+
⋯
+
n
2
x
n
=
2
n
+
2
,
x
1
+
2
2
x
2
+
3
2
x
3
+
⋯
+
n
2
x
n
=
n
2
+
n
+
4
,
x
1
+
2
3
x
2
+
3
3
x
3
+
⋯
+
n
3
x
n
=
n
3
+
n
+
8.
\begin{cases} \displaystyle x_1+ \phantom{2^2} x_2+ \phantom{3^2} x_3 + \cdots + \phantom{n^2} x_n &= n+2, \\ x_1 + 2\phantom{^2}x_2 + 3\phantom{^2}x_3 + \cdots + n\phantom{^2}x_n &= 2n+2, \\ x_1 + 2^2x_2 + 3^2 x_3 + \cdots + n^2x_n &= n^2 + n +4, \\ x_1+ 2^3x_2 + 3^3x_3+ \cdots + n^3x_n &= n^3 + n + 8. \end{cases}
⎩
⎨
⎧
x
1
+
2
2
x
2
+
3
2
x
3
+
⋯
+
n
2
x
n
x
1
+
2
2
x
2
+
3
2
x
3
+
⋯
+
n
2
x
n
x
1
+
2
2
x
2
+
3
2
x
3
+
⋯
+
n
2
x
n
x
1
+
2
3
x
2
+
3
3
x
3
+
⋯
+
n
3
x
n
=
n
+
2
,
=
2
n
+
2
,
=
n
2
+
n
+
4
,
=
n
3
+
n
+
8.
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