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Contests
International Contests
IMO Shortlist
2019 IMO Shortlist
A4
Pairwise distance-one products
Pairwise distance-one products
Source: IMO 2019 SL A4
September 22, 2020
algebra
IMO Shortlist
IMO Shortlist 2019
Problem Statement
Let
n
⩾
2
n\geqslant 2
n
⩾
2
be a positive integer and
a
1
,
a
2
,
…
,
a
n
a_1,a_2, \ldots ,a_n
a
1
,
a
2
,
…
,
a
n
be real numbers such that
a
1
+
a
2
+
⋯
+
a
n
=
0.
a_1+a_2+\dots+a_n=0.
a
1
+
a
2
+
⋯
+
a
n
=
0.
Define the set
A
A
A
by
A
=
{
(
i
,
j
)
∣
1
⩽
i
<
j
⩽
n
,
∣
a
i
−
a
j
∣
⩾
1
}
A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}
A
=
{
(
i
,
j
)
∣
1
⩽
i
<
j
⩽
n
,
∣
a
i
−
a
j
∣
⩾
1
}
Prove that, if
A
A
A
is not empty, then
∑
(
i
,
j
)
∈
A
a
i
a
j
<
0.
\sum_{(i, j) \in A} a_{i} a_{j}<0.
(
i
,
j
)
∈
A
∑
a
i
a
j
<
0.
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