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National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2024 Vietnam National Olympiad
5
5
Part of
2024 Vietnam National Olympiad
Problems
(1)
Polynomials having a large number of roots!
Source: 2024 Vietnam National Olympiad - Problem 5
1/6/2024
For each polynomial
P
(
x
)
P(x)
P
(
x
)
, define
P
1
(
x
)
=
P
(
x
)
,
∀
x
∈
R
,
P_1(x)=P(x), \forall x \in \mathbb{R},
P
1
(
x
)
=
P
(
x
)
,
∀
x
∈
R
,
P
2
(
x
)
=
P
(
P
1
(
x
)
)
,
∀
x
∈
R
,
P_2(x)=P(P_1(x)), \forall x \in \mathbb{R},
P
2
(
x
)
=
P
(
P
1
(
x
))
,
∀
x
∈
R
,
.
.
.
...
...
P
2024
(
x
)
=
P
(
P
2023
(
x
)
)
,
∀
x
∈
R
.
P_{2024}(x)=P(P_{2023}(x)), \forall x \in \mathbb{R}.
P
2024
(
x
)
=
P
(
P
2023
(
x
))
,
∀
x
∈
R
.
Let
a
>
2
a>2
a
>
2
be a real number. Is there a polynomial
P
P
P
with real coefficients such that for all
t
∈
(
−
a
,
a
)
t \in (-a, a)
t
∈
(
−
a
,
a
)
, the equation
P
2024
(
x
)
=
t
P_{2024}(x)=t
P
2024
(
x
)
=
t
has
2
2024
2^{2024}
2
2024
distinct real roots?
algebra
polynomial