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Bulgaria National Olympiad
2005 Bulgaria National Olympiad
6
6
Part of
2005 Bulgaria National Olympiad
Problems
(1)
Bulgaria 6
Source: BMO Problem 6
5/15/2005
Let
a
,
b
a,b
a
,
b
and
c
c
c
be positive integers such that
a
b
ab
ab
divides
c
(
c
2
−
c
+
1
)
c(c^{2}-c+1)
c
(
c
2
−
c
+
1
)
and
a
+
b
a+b
a
+
b
is divisible by
c
2
+
1
c^{2}+1
c
2
+
1
. Prove that the sets
{
a
,
b
}
\{a,b\}
{
a
,
b
}
and
{
c
,
c
2
−
c
+
1
}
\{c,c^{2}-c+1\}
{
c
,
c
2
−
c
+
1
}
coincide.
algebra
polynomial
symmetry
modular arithmetic
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number theory