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International Contests
Austrian-Polish
1996 Austrian-Polish Competition
6
6
Part of
1996 Austrian-Polish Competition
Problems
(1)
x_i^3(x_i^2 + x_{i+1}^2+... +x_{i+k-1}^2) = x_{i-1}^2 system
Source: Austrian - Polish 1996 APMC
5/3/2020
Given natural numbers
n
>
k
>
1
n > k > 1
n
>
k
>
1
, find all real solutions
x
1
,
.
.
.
,
x
n
x_1,..., x_n
x
1
,
...
,
x
n
of the system
x
i
3
(
x
i
2
+
x
i
+
1
2
+
.
.
.
+
x
i
+
k
−
1
2
)
=
x
i
−
1
2
x_i^3(x_i^2 + x_{i+1}^2+... +x_{i+k-1}^2) = x_{i-1}^2
x
i
3
(
x
i
2
+
x
i
+
1
2
+
...
+
x
i
+
k
−
1
2
)
=
x
i
−
1
2
for 1
≤
i
≤
n
\le i \le n
≤
i
≤
n
. Here
x
n
+
i
=
x
i
x_{n+i} = x_i
x
n
+
i
=
x
i
for all
i
i
i
.
system of equations
algebra