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Korea Contests
Korea National Olympiad
2006 Korea National Olympiad
3
Classic NT
Classic NT
Source: 2006 Korea National Olympiad #3
March 18, 2018
number theory
Problem Statement
For three positive integers
a
,
b
a,b
a
,
b
and
c
,
c,
c
,
if
gcd
(
a
,
b
,
c
)
=
1
\text{gcd}(a,b,c)=1
gcd
(
a
,
b
,
c
)
=
1
and
a
2
+
b
2
+
c
2
=
2
(
a
b
+
b
c
+
c
a
)
,
a^2+b^2+c^2=2(ab+bc+ca),
a
2
+
b
2
+
c
2
=
2
(
ab
+
b
c
+
c
a
)
,
prove that all of
a
,
b
,
c
a,b,c
a
,
b
,
c
is perfect square.
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