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US Math Competition Association
2021 USMCA
11
2021 USMCA National Championship #11
2021 USMCA National Championship #11
Source:
May 9, 2021
Problem Statement
Let
f
1
(
x
)
=
x
2
−
3
f_1 (x) = x^2 - 3
f
1
(
x
)
=
x
2
−
3
and
f
n
(
x
)
=
f
1
(
f
n
−
1
(
x
)
)
f_n (x) = f_1(f_{n-1} (x))
f
n
(
x
)
=
f
1
(
f
n
−
1
(
x
))
for
n
≥
2
n \ge 2
n
≥
2
. Let
m
n
m_n
m
n
be the smallest positive root of
f
n
f_n
f
n
, and
M
n
M_n
M
n
be the largest positive root of
f
n
f_n
f
n
. If
x
x
x
is the least number such that
M
n
≤
m
n
⋅
x
M_n \le m_n \cdot x
M
n
≤
m
n
⋅
x
for all
n
≥
1
n \ge 1
n
≥
1
, compute
x
2
x^2
x
2
.
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