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International Contests
Baltic Way
2017 Baltic Way
19
19
Part of
2017 Baltic Way
Problems
(1)
Number of carries
Source: Baltic Way 2017 Problem 19
11/11/2017
For an integer
n
≥
1
n\geq 1
n
≥
1
let
a
(
n
)
a(n)
a
(
n
)
denote the total number of carries which arise when adding
2017
2017
2017
and
n
⋅
2017
n\cdot 2017
n
⋅
2017
. The first few values are given by
a
(
1
)
=
1
a(1)=1
a
(
1
)
=
1
,
a
(
2
)
=
1
a(2)=1
a
(
2
)
=
1
,
a
(
3
)
=
0
a(3)=0
a
(
3
)
=
0
, which can be seen from the following: \begin{align*} 001 &&001 && 000 \\ 2017 &&4034 &&6051 \\ +2017 &&+2017 &&+2017\\ =4034 &&=6051 &&=8068\\ \end{align*} Prove that
a
(
1
)
+
a
(
2
)
+
.
.
.
+
a
(
1
0
2017
−
1
)
=
10
⋅
1
0
2017
−
1
9
.
a(1)+a(2)+...+a(10^{2017}-1)=10\cdot\frac{10^{2017}-1}{9}.
a
(
1
)
+
a
(
2
)
+
...
+
a
(
1
0
2017
−
1
)
=
10
⋅
9
1
0
2017
−
1
.
number theory