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Korea Contests
Korea National Olympiad
2021 Korea National Olympiad
P3
power of 2
power of 2
Source: 2021 kmo P3
November 14, 2021
number theory
Problem Statement
Show that for any positive integers
k
k
k
and
1
≤
a
≤
9
1 \leq a \leq 9
1
≤
a
≤
9
, there exists
n
n
n
such that satisfies the below statement. When
2
n
=
a
0
+
10
a
1
+
1
0
2
a
2
+
⋯
+
1
0
i
a
i
+
⋯
2^n=a_0+10a_1+10^2a_2+ \cdots +10^ia_i+ \cdots
2
n
=
a
0
+
10
a
1
+
1
0
2
a
2
+
⋯
+
1
0
i
a
i
+
⋯
(
0
≤
a
i
≤
9
(0 \leq a_i \leq 9
(
0
≤
a
i
≤
9
and
a
i
a_i
a
i
is integer),
a
k
a_k
a
k
is equal to
a
a
a
.
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