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IMC
2008 IMC
3
IMC 2008 Day 1 P3
IMC 2008 Day 1 P3
Source:
July 26, 2014
algebra
polynomial
quadratics
IMC
college contests
Problem Statement
Let
p
p
p
be a polynomial with integer coefficients and let
a
1
<
a
2
<
⋯
<
a
k
a_1<a_2<\cdots <a_k
a
1
<
a
2
<
⋯
<
a
k
be integers. Given that
p
(
a
i
)
≠
0
∀
i
=
1
,
2
,
⋯
,
k
p(a_i)\ne 0\forall\; i=1,2,\cdots, k
p
(
a
i
)
=
0∀
i
=
1
,
2
,
⋯
,
k
.(a) Prove
∃
a
∈
Z
\exists\; a\in \mathbb{Z}
∃
a
∈
Z
such that
p
(
a
i
)
∣
p
(
a
)
∀
i
=
1
,
2
,
…
,
k
p(a_i)\mid p(a)\;\;\forall i=1,2,\dots ,k
p
(
a
i
)
∣
p
(
a
)
∀
i
=
1
,
2
,
…
,
k
(b) Does there exist
a
∈
Z
a\in \mathbb{Z}
a
∈
Z
such that
∏
i
=
1
k
p
(
a
i
)
∣
p
(
a
)
\prod_{i=1}^{k}p(a_i)\mid p(a)
i
=
1
∏
k
p
(
a
i
)
∣
p
(
a
)
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