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9
2018 Algebra / NT #9
2018 Algebra / NT #9
Source:
February 12, 2018
inequalities
algebra
Problem Statement
Assume the quartic
x
4
−
a
x
3
+
b
x
2
−
a
x
+
d
=
0
x^4-ax^3+bx^2-ax+d=0
x
4
−
a
x
3
+
b
x
2
−
a
x
+
d
=
0
has four real roots
1
2
≤
x
1
,
x
2
,
x
3
,
x
4
≤
2.
\frac{1}{2}\leq x_1,x_2,x_3,x_4\leq 2.
2
1
≤
x
1
,
x
2
,
x
3
,
x
4
≤
2.
Find the maximum possible value of
(
x
1
+
x
2
)
(
x
1
+
x
3
)
x
4
(
x
4
+
x
2
)
(
x
4
+
x
3
)
x
1
.
\frac{(x_1+x_2)(x_1+x_3)x_4}{(x_4+x_2)(x_4+x_3)x_1}.
(
x
4
+
x
2
)
(
x
4
+
x
3
)
x
1
(
x
1
+
x
2
)
(
x
1
+
x
3
)
x
4
.
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