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Problems
Contests
National and Regional Contests
PEN Problems
PEN I Problems
14
14
Part of
PEN I Problems
Problems
(1)
I 14
Source:
5/25/2007
Let
a
,
b
,
n
a, b, n
a
,
b
,
n
be positive integers with
gcd
(
a
,
b
)
=
1
\gcd(a, b)=1
g
cd
(
a
,
b
)
=
1
. Prove that
∑
k
{
a
k
+
b
n
}
=
n
−
1
2
,
\sum_{k}\left\{ \frac{ak+b}{n}\right\}=\frac{n-1}{2},
k
∑
{
n
ak
+
b
}
=
2
n
−
1
,
where
k
k
k
runs through a complete system of residues modulo
m
m
m
.