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Austrian-Polish
2005 Austrian-Polish Competition
7
7
Part of
2005 Austrian-Polish Competition
Problems
(1)
System of equations, n variables, n^3-n^2
Source: Austrian-Polish 2005, Problem 7
7/5/2015
For each natural number
n
≥
2
n\geq 2
n
≥
2
, solve the following system of equations in the integers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ..., x_n
x
1
,
x
2
,
...
,
x
n
:
(
n
2
−
n
)
x
i
+
(
∏
j
≠
i
x
j
)
S
=
n
3
−
n
2
,
∀
1
≤
i
≤
n
(n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n
(
n
2
−
n
)
x
i
+
j
=
i
∏
x
j
S
=
n
3
−
n
2
,
∀1
≤
i
≤
n
where
S
=
x
1
2
+
x
2
2
+
⋯
+
x
n
2
.
S=x_1^2+x_2^2+\dots+x_n^2.
S
=
x
1
2
+
x
2
2
+
⋯
+
x
n
2
.
algebra
system of equations
number theory