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National and Regional Contests
Mexico Contests
Mexico National Olympiad
2010 Mexico National Olympiad
3
(pq)^r+(qr)^p+(rp)^q
(pq)^r+(qr)^p+(rp)^q
Source: OMM 2010 6
July 15, 2014
number theory unsolved
number theory
Problem Statement
Let
p
p
p
,
q
q
q
, and
r
r
r
be distinct positive prime numbers. Show that if
p
q
r
∣
(
p
q
)
r
+
(
q
r
)
p
+
(
r
p
)
q
−
1
,
pqr\mid (pq)^r+(qr)^p+(rp)^q-1,
pq
r
∣
(
pq
)
r
+
(
q
r
)
p
+
(
r
p
)
q
−
1
,
then
(
p
q
r
)
3
∣
3
(
(
p
q
)
r
+
(
q
r
)
p
+
(
r
p
)
q
−
1
)
.
(pqr)^3\mid 3((pq)^r+(qr)^p+(rp)^q-1).
(
pq
r
)
3
∣
3
((
pq
)
r
+
(
q
r
)
p
+
(
r
p
)
q
−
1
)
.
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