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2020 Princeton University Math Competition
12
12
Part of
2020 Princeton University Math Competition
Problems
(1)
2020 PUMaC Team 12
Source:
1/1/2022
Given a sequence
a
0
,
a
1
,
a
2
,
.
.
.
,
a
n
a_0, a_1, a_2, ... , a_n
a
0
,
a
1
,
a
2
,
...
,
a
n
, let its arithmetic approximant be the arithmetic sequence
b
0
,
b
1
,
.
.
.
,
b
n
b_0, b_1, ... , b_n
b
0
,
b
1
,
...
,
b
n
that minimizes the quantity
∑
i
=
0
n
(
b
i
−
a
i
)
2
\sum_{i=0}^{n}(b_i -a_i)^2
∑
i
=
0
n
(
b
i
−
a
i
)
2
, and denote this quantity the sequence’s anti-arithmeticity. Denote the number of integer sequences whose arithmetic approximant is the sequence
4
4
4
,
8
8
8
,
12
12
12
,
16
16
16
and whose anti-arithmeticity is at most
20
20
20
.
algebra