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239 Open Math Olympiad
2018 239 Open Mathematical Olympiad
10-11.3
10-11.3
Part of
2018 239 Open Mathematical Olympiad
Problems
(1)
239 Open MO, 2018, Senior League, Problem 3
Source: 239 Open MO, 2018, Senior League, Problem 3
4/4/2023
Given a prime number
p
>
5
p>5
p
>
5
. It is known that the length of the smallest period of the fraction
1
/
p
1/p
1/
p
is a multiple of three. This period (including possible leading zeros) was written on a strip of paper and cut into three equal-length parts
a
a
a
,
b
b
b
,
c
c
c
(they may also have leading zeros). What could be the sum of the three periodic fractions:
0.
(
a
)
0.(a)
0.
(
a
)
,
0.
(
b
)
0.(b)
0.
(
b
)
, and
0.
(
c
)
0.(c)
0.
(
c
)
?Proposed by A. Khrabrov
number theory