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National and Regional Contests
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Canadian Open Math Challenge
2017 Canadian Open Math Challenge
C1
C1
Part of
2017 Canadian Open Math Challenge
Problems
(1)
2017 COMC C1
Source:
10/12/2018
Source: 2017 Canadian Open Math Challenge, Problem C1 —-- For a positive integer
n
n
n
, we define function
P
(
n
)
P(n)
P
(
n
)
to be the sum of the digits of
n
n
n
plus the number of digits of
n
n
n
. For example,
P
(
45
)
=
4
+
5
+
2
=
11
P(45) = 4 + 5 + 2 = 11
P
(
45
)
=
4
+
5
+
2
=
11
. (Note that the first digit of
n
n
n
reading from left to right, cannot be
0
0
0
).
\qquad
(a) Determine
P
(
2017
)
P(2017)
P
(
2017
)
.
\qquad
(b) Determine all numbers
n
n
n
such that
P
(
n
)
=
4
P(n) = 4
P
(
n
)
=
4
.
\qquad
(c) Determine with an explanation whether there exists a number
n
n
n
for which
P
(
n
)
ā
P
(
n
+
1
)
>
50
P(n) - P(n + 1) > 50
P
(
n
)
ā
P
(
n
+
1
)
>
50
.
Comc
2017 COMC