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Problems
Contests
International Contests
IMO Longlists
1983 IMO Longlists
26
26
Part of
1983 IMO Longlists
Problems
(1)
Show that 2abc-ab-bc-ca cannot be represented in such form
Source:
10/5/2010
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive integers satisfying
gcd
(
a
,
b
)
=
gcd
(
b
,
c
)
=
gcd
(
c
,
a
)
=
1
\gcd (a, b) = \gcd (b, c) = \gcd (c, a) = 1
g
cd
(
a
,
b
)
=
g
cd
(
b
,
c
)
=
g
cd
(
c
,
a
)
=
1
. Show that
2
a
b
c
−
a
b
−
b
c
−
c
a
2abc-ab-bc-ca
2
ab
c
−
ab
−
b
c
−
c
a
cannot be represented as
b
c
x
+
c
a
y
+
a
b
z
bcx+cay +abz
b
c
x
+
c
a
y
+
ab
z
with nonnegative integers
x
,
y
,
z
.
x, y, z.
x
,
y
,
z
.
number theory unsolved
number theory