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2017 MMATHS
3
3
Part of
2017 MMATHS
Problems
(1)
2017 MMATHS Tiebreaker p3 - f^{((2017^{2017})!)(q)} \ne (q + 2017)(qP(q) - 1)
Source:
10/8/2023
Let
f
:
R
→
R
f : R \to R
f
:
R
→
R
, and let
P
P
P
be a nonzero polynomial with degree no more than
2015
2015
2015
. For any nonnegative integer
n
n
n
,
f
(
n
)
(
x
)
f^{(n)}(x)
f
(
n
)
(
x
)
denotes the function defined as
f
f
f
composed with itself
n
n
n
times. For example,
f
(
0
)
(
x
)
=
x
f^{(0)}(x) = x
f
(
0
)
(
x
)
=
x
,
f
(
1
)
(
x
)
=
f
(
x
)
f^{(1)}(x) = f(x)
f
(
1
)
(
x
)
=
f
(
x
)
,
f
(
2
)
(
x
)
=
f
(
f
(
x
)
)
f^{(2)}(x) = f(f(x))
f
(
2
)
(
x
)
=
f
(
f
(
x
))
, etc. Show that there always exists a real number
q
q
q
such that
f
(
(
201
7
2017
)
!
)
(
q
)
≠
(
q
+
2017
)
(
q
P
(
q
)
−
1
)
.
f^{((2017^{2017})!)(q)} \ne (q + 2017)(qP(q) - 1).
f
((
201
7
2017
)!)
(
q
)
=
(
q
+
2017
)
(
qP
(
q
)
−
1
)
.
algebra
polynomial
MMATHS