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Ecuador Mathematical Olympiad (OMEC)
2016 Ecuador NMO (OMEC)
6
6
Part of
2016 Ecuador NMO (OMEC)
Problems
(1)
sum c_ix_i is not divisible by n^3 2016 Ecuador NMO (OMEC) 3.6
Source:
9/17/2021
A positive integer
n
n
n
is "olympic" if there are
n
n
n
non-negative integers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ..., x_n
x
1
,
x
2
,
...
,
x
n
that satisfy that:
∙
\bullet
∙
There is at least one positive integer
j
j
j
:
1
≤
j
≤
n
1 \le j \le n
1
≤
j
≤
n
such that
x
j
≠
0
x_j \ne 0
x
j
=
0
.
∙
\bullet
∙
For any way of choosing
n
n
n
numbers
c
1
,
c
2
,
.
.
.
,
c
n
c_1, c_2, ..., c_n
c
1
,
c
2
,
...
,
c
n
from the set
{
−
1
,
0
,
1
}
\{-1, 0, 1\}
{
−
1
,
0
,
1
}
, where not all
c
i
c_i
c
i
are equal to zero, it is true that the sum
c
1
x
1
+
c
2
x
2
+
.
.
.
+
c
n
x
n
c_1x_1 + c_2x_2 +... + c_nx_n
c
1
x
1
+
c
2
x
2
+
...
+
c
n
x
n
is not divisible by
n
3
n^3
n
3
. Find the largest positive "olympic" integer.
number theory
divisible