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APMO
2000 APMO
2
Equal sums
Equal sums
Source: APMO 2000
April 1, 2006
combinatorics unsolved
combinatorics
Problem Statement
Find all permutations
a
1
,
a
2
,
…
,
a
9
a_1, a_2, \ldots, a_9
a
1
,
a
2
,
…
,
a
9
of
1
,
2
,
…
,
9
1, 2, \ldots, 9
1
,
2
,
…
,
9
such that
a
1
+
a
2
+
a
3
+
a
4
=
a
4
+
a
5
+
a
6
+
a
7
=
a
7
+
a
8
+
a
9
+
a
1
a_1+a_2+a_3+a_4=a_4+a_5+a_6+a_7= a_7+a_8+a_9+a_1
a
1
+
a
2
+
a
3
+
a
4
=
a
4
+
a
5
+
a
6
+
a
7
=
a
7
+
a
8
+
a
9
+
a
1
and
a
1
2
+
a
2
2
+
a
3
2
+
a
4
2
=
a
4
2
+
a
5
2
+
a
6
2
+
a
7
2
=
a
7
2
+
a
8
2
+
a
9
2
+
a
1
2
a_1^2+a_2^2+a_3^2+a_4^2=a_4^2+a_5^2+a_6^2+a_7^2= a_7^2+a_8^2+a_9^2+a_1^2
a
1
2
+
a
2
2
+
a
3
2
+
a
4
2
=
a
4
2
+
a
5
2
+
a
6
2
+
a
7
2
=
a
7
2
+
a
8
2
+
a
9
2
+
a
1
2
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