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2010 Iran MO (2nd Round)
4
Iran National Math Olympiad-Day 2-Problem 4
Iran National Math Olympiad-Day 2-Problem 4
Source:
April 30, 2010
algebra
polynomial
inequalities
algebra unsolved
Problem Statement
Let
P
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
P(x)=ax^3+bx^2+cx+d
P
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
be a polynomial with real coefficients such that
min
{
d
,
b
+
d
}
>
max
{
∣
c
∣
,
∣
a
+
c
∣
}
\min\{d,b+d\}> \max\{|{c}|,|{a+c}|\}
min
{
d
,
b
+
d
}
>
max
{
∣
c
∣
,
∣
a
+
c
∣
}
Prove that
P
(
x
)
P(x)
P
(
x
)
do not have a real root in
[
−
1
,
1
]
[-1,1]
[
−
1
,
1
]
.
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