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2021 Science ON
2021 Science ON grade VIII
4
Many terms, unusual inequality
Many terms, unusual inequality
Source: Science ON 2021 grade VIII/4
March 8, 2021
Inequality
inequalities
algebra
Problem Statement
Consider positive real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
. Prove the inequality
1
x
+
1
y
+
1
z
+
9
x
+
y
+
z
≥
3
(
(
1
2
x
+
y
+
1
x
+
2
y
)
+
(
1
2
y
+
z
+
1
y
+
2
z
)
+
(
1
2
z
+
x
+
1
x
+
2
z
)
)
.
\frac 1x+\frac 1y+\frac 1z+\frac{9}{x+y+z}\ge 3\left (\left (\frac{1}{2x+y}+\frac{1}{x+2y}\right )+\left (\frac{1}{2y+z}+\frac{1}{y+2z}\right )+\left (\frac{1}{2z+x}+\frac{1}{x+2z}\right )\right ).
x
1
+
y
1
+
z
1
+
x
+
y
+
z
9
≥
3
(
(
2
x
+
y
1
+
x
+
2
y
1
)
+
(
2
y
+
z
1
+
y
+
2
z
1
)
+
(
2
z
+
x
1
+
x
+
2
z
1
)
)
.
(Vlad Robu \& Sergiu Novac)
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