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National and Regional Contests
Russia Contests
239 Open Math Olympiad
2017 239 Open Mathematical Olympiad
4
f^k(a)-a has no root
f^k(a)-a has no root
Source: 239 2017 S4
May 22, 2020
algebra
polynomial
Problem Statement
A polynomial
f
(
x
)
f(x)
f
(
x
)
with integer coefficients is given. We define
d
(
a
,
k
)
=
∣
f
k
(
a
)
−
a
∣
.
d(a,k)=|f^k(a)-a|.
d
(
a
,
k
)
=
∣
f
k
(
a
)
−
a
∣.
It is known that for each integer
a
a
a
and natural number
k
k
k
,
d
(
a
,
k
)
d(a,k)
d
(
a
,
k
)
is positive. Prove that for all such
a
,
k
a,k
a
,
k
,
d
(
a
,
k
)
≥
k
3
.
d(a,k) \geq \frac{k}{3}.
d
(
a
,
k
)
≥
3
k
.
(
f
k
(
x
)
=
f
(
f
k
−
1
(
x
)
)
,
f
0
(
x
)
=
x
.
f^k(x)=f(f^{k-1}(x)), f^0(x)=x.
f
k
(
x
)
=
f
(
f
k
−
1
(
x
))
,
f
0
(
x
)
=
x
.
)
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