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KoMaL A Problems
KoMaL A Problems 2023/2024
A. 867
A. 867
Part of
KoMaL A Problems 2023/2024
Problems
(1)
Sum of absoulte values of polynomial at roots of the other
Source: KoMaL A. 867
1/13/2024
Let
p
(
x
)
p(x)
p
(
x
)
be a monic integer polynomial of degree
n
n
n
that has
n
n
n
real roots,
α
1
,
α
2
,
…
,
α
n
\alpha_1,\alpha_2,\ldots, \alpha_n
α
1
,
α
2
,
…
,
α
n
. Let
q
(
x
)
q(x)
q
(
x
)
be an arbitrary integer polynomial that is relatively prime to polynomial
p
(
x
)
p(x)
p
(
x
)
. Prove that
∑
i
=
1
n
∣
q
(
α
i
)
∣
≥
n
.
\sum_{i=1}^n \left|q(\alpha_i)\right|\ge n.
i
=
1
∑
n
∣
q
(
α
i
)
∣
≥
n
.
Submitted by Dávid Matolcsi, Berkeley
algebra
polynomial