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International Contests
IMO Longlists
1984 IMO Longlists
56
56
Part of
1984 IMO Longlists
Problems
(1)
Finding three nonnegative integers...
Source:
10/13/2010
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be nonnegative integers such that
a
≤
b
≤
c
,
2
b
≠
a
+
c
a \le b \le c, 2b \neq a + c
a
≤
b
≤
c
,
2
b
=
a
+
c
and
a
+
b
+
c
3
\frac{a+b+c}{3}
3
a
+
b
+
c
is an integer. Is it possible to find three nonnegative integers
d
,
e
d, e
d
,
e
, and
f
f
f
such that
d
≤
e
≤
f
,
f
≠
c
d \le e \le f, f \neq c
d
≤
e
≤
f
,
f
=
c
, and such that
a
2
+
b
2
+
c
2
=
d
2
+
e
2
+
f
2
a^2+b^2+c^2 = d^2 + e^2 + f^2
a
2
+
b
2
+
c
2
=
d
2
+
e
2
+
f
2
?
number theory unsolved
number theory