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International Contests
Austrian-Polish
1984 Austrian-Polish Competition
5
a_nx_1 + a_1x_2 +...-+ a_{n-1}x_n = yx_n , system
a_nx_1 + a_1x_2 +...-+ a_{n-1}x_n = yx_n , system
Source: Austrian Polish 1984 APMC
April 30, 2020
system of equations
algebra
Problem Statement
Given
n
>
2
n > 2
n
>
2
nonnegative distinct integers
a
1
,
.
.
.
,
a
n
a_1,...,a_n
a
1
,
...
,
a
n
, find all nonnegative integers
y
y
y
and
x
1
,
.
.
.
,
x
n
x_1,...,x_n
x
1
,
...
,
x
n
satisfying
g
c
d
(
x
1
,
.
.
.
,
x
n
)
=
1
gcd(x_1,...,x_n) = 1
g
c
d
(
x
1
,
...
,
x
n
)
=
1
and
{
a
1
x
1
+
a
2
x
2
+
.
.
.
+
a
n
x
n
=
y
x
1
a
2
x
1
+
a
3
x
2
+
.
.
.
+
a
1
x
n
=
y
x
2
.
.
.
a
n
x
1
+
a
1
x
2
+
.
.
.
+
a
n
−
1
x
n
=
y
x
n
\begin{cases} a_1x_1 + a_2x_2 +...+ a_nx_n = yx_1 \\ a_2x_1 + a_3x_2 +...+ a_1x_n = yx_2 \\ ... \\ a_nx_1 + a_1x_2 +...+ a_{n-1}x_n = yx_n \end{cases}
⎩
⎨
⎧
a
1
x
1
+
a
2
x
2
+
...
+
a
n
x
n
=
y
x
1
a
2
x
1
+
a
3
x
2
+
...
+
a
1
x
n
=
y
x
2
...
a
n
x
1
+
a
1
x
2
+
...
+
a
n
−
1
x
n
=
y
x
n
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