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All-Russian Olympiad
2007 All-Russian Olympiad
2
inequality on polynomials
inequality on polynomials
Source: All-Russian 2007
May 4, 2007
inequalities
algebra
polynomial
induction
algebra unsolved
Problem Statement
Given polynomial
P
(
x
)
=
a
0
x
n
+
a
1
x
n
−
1
+
⋯
+
a
n
−
1
x
+
a
n
P(x) = a_{0}x^{n}+a_{1}x^{n-1}+\dots+a_{n-1}x+a_{n}
P
(
x
)
=
a
0
x
n
+
a
1
x
n
−
1
+
⋯
+
a
n
−
1
x
+
a
n
. Put
m
=
min
{
a
0
,
a
0
+
a
1
,
…
,
a
0
+
a
1
+
⋯
+
a
n
}
m=\min \{ a_{0}, a_{0}+a_{1}, \dots, a_{0}+a_{1}+\dots+a_{n}\}
m
=
min
{
a
0
,
a
0
+
a
1
,
…
,
a
0
+
a
1
+
⋯
+
a
n
}
. Prove that
P
(
x
)
≥
m
x
n
P(x) \ge mx^{n}
P
(
x
)
≥
m
x
n
for
x
≥
1
x \ge 1
x
≥
1
. A. Khrabrov
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