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National and Regional Contests
Italy Contests
ITAMO
2018 ITAMO
5
5
Part of
2018 ITAMO
Problems
(1)
ITAMO problem 5
Source: ITAMO 2018
5/9/2018
5.
5.
5.
Let x be a real number with
0
<
x
<
1
0<x<1
0
<
x
<
1
and let
0.
c
1
c
2
c
3
.
.
.
0.c_1c_2c_3...
0.
c
1
c
2
c
3
...
be the decimal expansion of x.Denote by
B
(
x
)
B(x)
B
(
x
)
the set of all subsequences of
c
1
c
2
c
3
c_1c_2c_3
c
1
c
2
c
3
that consist of 6 consecutive digits. For instance ,
B
(
1
22
)
=
045454
,
454545
,
545454
B(\frac{1}{22})={045454,454545,545454}
B
(
22
1
)
=
045454
,
454545
,
545454
Find the minimum number of elements of
B
(
x
)
B(x)
B
(
x
)
as
x
x
x
varies among all irrational numbers with
0
<
x
<
1
0<x<1
0
<
x
<
1
combinatorics