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IMC
2009 IMC
4
IMC 2009 Day 2 P4
IMC 2009 Day 2 P4
Source:
July 17, 2014
modular arithmetic
algebra
polynomial
IMC
college contests
Problem Statement
Let
p
p
p
be a prime number and
W
⊆
F
p
[
x
]
\mathbf{W}\subseteq \mathbb{F}_p[x]
W
⊆
F
p
[
x
]
be the smallest set satisfying the following :(a)
x
+
1
∈
W
x+1\in \mathbf{W}
x
+
1
∈
W
and
x
p
−
2
+
x
p
−
3
+
⋯
+
x
2
+
2
x
+
1
∈
W
x^{p-2}+x^{p-3}+\cdots +x^2+2x+1\in \mathbf{W}
x
p
−
2
+
x
p
−
3
+
⋯
+
x
2
+
2
x
+
1
∈
W
(b) For
γ
1
,
γ
2
\gamma_1,\gamma_2
γ
1
,
γ
2
in
W
\mathbf{W}
W
, we also have
γ
(
x
)
∈
W
\gamma(x)\in \mathbf{W}
γ
(
x
)
∈
W
, where
γ
(
x
)
\gamma(x)
γ
(
x
)
is the remainder
(
γ
1
∘
γ
2
)
(
x
)
(
m
o
d
x
p
−
x
)
(\gamma_1\circ \gamma_2)(x)\pmod {x^p-x}
(
γ
1
∘
γ
2
)
(
x
)
(
mod
x
p
−
x
)
. How many polynomials are in
W
?
\mathbf{W}?
W
?
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