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2022 South East Mathematical Olympiad
1
China South East Mathematical Olympiad 2022 Grade11 Q1
China South East Mathematical Olympiad 2022 Grade11 Q1
Source: Jiangxi jian
August 2, 2022
inequalities
algebra
Problem Statement
Let
x
1
,
x
2
,
x
3
x_1,x_2,x_3
x
1
,
x
2
,
x
3
be three positive real roots of the equation
x
3
+
a
x
2
+
b
x
+
c
=
0
x^3+ax^2+bx+c=0
x
3
+
a
x
2
+
b
x
+
c
=
0
(
a
,
b
,
c
∈
R
)
(a,b,c\in R)
(
a
,
b
,
c
∈
R
)
and
x
1
+
x
2
+
x
3
≤
1.
x_1+x_2+x_3\leq 1.
x
1
+
x
2
+
x
3
≤
1.
Prove that
a
3
(
1
+
a
+
b
)
−
9
c
(
3
+
3
a
+
a
2
)
≤
0
a^3(1+a+b)-9c(3+3a+a^2)\leq 0
a
3
(
1
+
a
+
b
)
−
9
c
(
3
+
3
a
+
a
2
)
≤
0
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