MathDB
Problems
Contests
National and Regional Contests
PEN Problems
PEN E Problems
32
32
Part of
PEN E Problems
Problems
(1)
E 32
Source:
5/25/2007
Let
n
≥
5
n \ge 5
n
≥
5
be an integer. Show that
n
n
n
is prime if and only if
n
i
n
j
≠
n
p
n
q
n_{i} n_{j} \neq n_{p} n_{q}
n
i
n
j
=
n
p
n
q
for every partition of
n
n
n
into
4
4
4
integers,
n
=
n
1
+
n
2
+
n
3
+
n
4
n=n_{1}+n_{2}+n_{3}+n_{4}
n
=
n
1
+
n
2
+
n
3
+
n
4
, and for each permutation
(
i
,
j
,
p
,
q
)
(i, j, p, q)
(
i
,
j
,
p
,
q
)
of
(
1
,
2
,
3
,
4
)
(1, 2, 3, 4)
(
1
,
2
,
3
,
4
)
.
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