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Sweden Contests
Swedish Mathematical Competition
1997 Swedish Mathematical Competition
5
5
Part of
1997 Swedish Mathematical Competition
Problems
(1)
s(k)+s(f(n)-k) = n, sum of digits
Source: 1997 Swedish Mathematical Competition p5
4/2/2021
Let
s
(
m
)
s(m)
s
(
m
)
denote the sum of (decimal) digits of a positive integer
m
m
m
. Prove that for every integer
n
>
1
n > 1
n
>
1
not equal to
10
10
10
there is a unique integer
f
(
n
)
≥
2
f(n) \ge 2
f
(
n
)
≥
2
such that
s
(
k
)
+
s
(
f
(
n
)
−
k
)
=
n
s(k)+s(f(n)-k) = n
s
(
k
)
+
s
(
f
(
n
)
−
k
)
=
n
for all integers
k
k
k
with
0
<
k
<
f
(
n
)
0 < k < f(n)
0
<
k
<
f
(
n
)
.
number theory
sum of digits