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Kyrgyzstan Contests
Kyrgyzstan National Olympiad
2010 Kyrgyzstan National Olympiad
7
divisible triples(KgNM2010)
divisible triples(KgNM2010)
Source:
April 25, 2011
number theory unsolved
number theory
Problem Statement
Find all natural triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
, such that:
a
−
)
a
≤
b
≤
c
a - )\,a \le b \le c
a
−
)
a
≤
b
≤
c
b
−
)
(
a
,
b
,
c
)
=
1
b - )\,(a,b,c) = 1
b
−
)
(
a
,
b
,
c
)
=
1
c
−
)
a
2
b
∣
a
3
+
b
3
+
c
3
,
b
2
c
∣
a
3
+
b
3
+
c
3
,
c
2
a
∣
a
3
+
b
3
+
c
3
c - )\,\left. {{a^2}b} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{b^2}c} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{c^2}a} \right|{a^3} + {b^3} + {c^3}
c
−
)
a
2
b
a
3
+
b
3
+
c
3
,
b
2
c
a
3
+
b
3
+
c
3
,
c
2
a
a
3
+
b
3
+
c
3
.
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