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1988 IMO Longlists
60
60
Part of
1988 IMO Longlists
Problems
(1)
Summed up product divisible by 1001
Source: IMO LongList 1988, Poland 3, Problem 60 of ILL
11/3/2005
Given integers
a
1
,
…
,
a
10
,
a_1, \ldots, a_{10},
a
1
,
…
,
a
10
,
prove that there exist a non-zero sequence
{
x
1
,
…
,
x
10
}
\{x_1, \ldots, x_{10}\}
{
x
1
,
…
,
x
10
}
such that all
x
i
x_i
x
i
belong to
{
−
1
,
0
,
1
}
\{-1,0,1\}
{
−
1
,
0
,
1
}
and the number
∑
i
=
1
10
x
i
⋅
a
i
\sum^{10}_{i=1} x_i \cdot a_i
∑
i
=
1
10
x
i
⋅
a
i
is divisible by 1001.
number theory unsolved
number theory