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Taiwan Contests
Taiwan APMO Prelininary
2021 Taiwan APMO Preliminary First Round
6
6
Part of
2021 Taiwan APMO Preliminary First Round
Problems
(1)
2AB=\overline{AB}
Source: 2021 Taiwan APMO Preliminary First Round
11/21/2020
Find all positive integers
A
,
B
A,B
A
,
B
satisfying the following properties: (i)
A
A
A
and
B
B
B
has same digit in decimal. (ii)
2
⋅
A
⋅
B
=
A
B
‾
2\cdot A\cdot B=\overline{AB}
2
⋅
A
⋅
B
=
A
B
(Here
⋅
\cdot
⋅
denotes multiplication,
A
B
‾
\overline{AB}
A
B
denotes we write
A
A
A
and
B
B
B
in turn. For example, if
A
=
12
,
B
=
34
A=12,B=34
A
=
12
,
B
=
34
, then
A
B
‾
=
1234
\overline{AB}=1234
A
B
=
1234
)
number theory