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2008 JBMO Shortlist
8
Σx_i \cdot Σ1/x_i >= 4 (Σχ_i/(x_ix_j+1))^3 , i=1,2,3
Σx_i \cdot Σ1/x_i >= 4 (Σχ_i/(x_ix_j+1))^3 , i=1,2,3
Source: JBMO 2008 Shortlist A8
October 14, 2017
JBMO
inequalities
algebra
Problem Statement
Show that
(
x
+
y
+
z
)
(
1
x
+
1
y
+
1
z
)
≥
4
(
x
x
y
+
1
+
y
y
z
+
1
+
z
z
x
+
1
)
2
(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\big)^2
(
x
+
y
+
z
)
(
x
1
+
y
1
+
z
1
)
≥
4
(
x
y
+
1
x
+
yz
+
1
y
+
z
x
+
1
z
)
2
, for all real positive numbers
x
,
y
x, y
x
,
y
and
z
z
z
.
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