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CNCM Online Round 1
3
3
Part of
CNCM Online Round 1
Problems
(1)
CNCM Online R1P3
Source:
9/2/2020
Define
S
(
N
)
S(N)
S
(
N
)
to be the sum of the digits of
N
N
N
when it is written in base
10
10
10
, and take
S
k
(
N
)
=
S
(
S
(
…
(
N
)
…
)
)
S^k(N) = S(S(\dots(N)\dots))
S
k
(
N
)
=
S
(
S
(
…
(
N
)
…
))
with
k
k
k
applications of
S
S
S
. The
stability
of a number
N
N
N
is defined to be the smallest positive integer
K
K
K
where
S
K
(
N
)
=
S
K
+
1
(
N
)
=
S
K
+
2
(
N
)
=
…
S^K(N) = S^{K+1}(N) = S^{K+2}(N) = \dots
S
K
(
N
)
=
S
K
+
1
(
N
)
=
S
K
+
2
(
N
)
=
…
. Let
T
3
T_3
T
3
be the set of all natural numbers with stability
3
3
3
. Compute the sum of the two least entries of
T
3
T_3
T
3
. Proposed by Albert Wang (awang11)