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CNCM Online Round 2
5
5
Part of
CNCM Online Round 2
Problems
(1)
CNCM Online R2P5
Source:
8/8/2020
Consider a regular
n
n
n
-gon of side length 1. For each of its vertices, a circle of radius one is drawn centered at that vertex. The resulting figure, consisting of the polygon and the
n
n
n
circles, partitions the plane into
f
(
n
)
f(n)
f
(
n
)
finite, bounded regions. Find
∑
n
=
3
25
f
(
n
)
.
\sum_{n=3}^{25} f(n).
n
=
3
∑
25
f
(
n
)
.
The first term corresponding to
i
=
3
i=3
i
=
3
is shown; each of the various colors corresponds to a distinct region with
f
(
3
)
=
10
f(3)=10
f
(
3
)
=
10
. Note that the lines corresponding to the polygon are treated no differently than the arcs corresponding to the circles in counting regions. Proposed by Hari Desikan (HariDesikan)
CNCM
Problem Discussion