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National and Regional Contests
PEN Problems
PEN S Problems
18
18
Part of
PEN S Problems
Problems
(1)
S 18
Source:
5/25/2007
Denote by
S
S
S
the set of all primes
p
p
p
such that the decimal representation of
1
p
\frac{1}{p}
p
1
has the fundamental period of divisible by
3
3
3
. For every
p
∈
S
p \in S
p
∈
S
such that
1
p
\frac{1}{p}
p
1
has the fundamental period
3
r
3r
3
r
one may write
1
p
=
0.
a
1
a
2
⋯
a
3
r
a
1
a
2
⋯
a
3
r
⋯
,
\frac{1}{p}= 0.a_{1}a_{2}\cdots a_{3r}a_{1}a_{2}\cdots a_{3r}\cdots,
p
1
=
0.
a
1
a
2
⋯
a
3
r
a
1
a
2
⋯
a
3
r
⋯
,
where
r
=
r
(
p
)
r=r(p)
r
=
r
(
p
)
. For every
p
∈
S
p \in S
p
∈
S
and every integer
k
≥
1
k \ge 1
k
≥
1
define
f
(
k
,
p
)
=
a
k
+
a
k
+
r
(
p
)
+
a
k
+
2
r
(
p
)
.
f(k, p)=a_{k}+a_{k+r(p)}+a_{k+2r(p)}.
f
(
k
,
p
)
=
a
k
+
a
k
+
r
(
p
)
+
a
k
+
2
r
(
p
)
.
[*] Prove that
S
S
S
is finite. [*] Find the highest value of
f
(
k
,
p
)
f(k, p)
f
(
k
,
p
)
for
k
≥
1
k \ge 1
k
≥
1
and
p
∈
S
p \in S
p
∈
S
.
Miscellaneous Problems