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National and Regional Contests
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Indonesia MO
2019 Indonesia MO
8
Annoying C
Annoying C
Source: INAMO 2019 P8
July 3, 2019
combinatorics
Problem Statement
Let
n
>
1
n > 1
n
>
1
be a positive integer and
a
1
,
a
2
,
…
,
a
2
n
∈
{
−
n
,
−
n
+
1
,
…
,
n
−
1
,
n
}
a_1, a_2, \dots, a_{2n} \in \{ -n, -n + 1, \dots, n - 1, n \}
a
1
,
a
2
,
…
,
a
2
n
∈
{
−
n
,
−
n
+
1
,
…
,
n
−
1
,
n
}
. Suppose
a
1
+
a
2
+
a
3
+
⋯
+
a
2
n
=
n
+
1
a_1 + a_2 + a_3 + \dots + a_{2n} = n + 1
a
1
+
a
2
+
a
3
+
⋯
+
a
2
n
=
n
+
1
Prove that some of
a
1
,
a
2
,
…
,
a
2
n
a_1, a_2, \dots, a_{2n}
a
1
,
a
2
,
…
,
a
2
n
have sum 0.
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