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Problems
Contests
National and Regional Contests
Israel Contests
Israel National Olympiad
2022 Israel National Olympiad
P6
P6
Part of
2022 Israel National Olympiad
Problems
(1)
Inequality with square roots
Source: 2022 Israel National Olympiad P6
12/16/2022
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be non-negative real numbers. Prove that:
(
2
x
+
y
)
(
2
x
+
z
)
+
(
2
y
+
x
)
(
2
y
+
z
)
+
(
2
z
+
x
)
(
2
z
+
y
)
≥
\sqrt{(2x+y)(2x+z)}+\sqrt{(2y+x)(2y+z)}+\sqrt{(2z+x)(2z+y)}\geq
(
2
x
+
y
)
(
2
x
+
z
)
+
(
2
y
+
x
)
(
2
y
+
z
)
+
(
2
z
+
x
)
(
2
z
+
y
)
≥
≥
(
x
+
2
y
)
(
x
+
2
z
)
+
(
y
+
2
x
)
(
y
+
2
z
)
+
(
z
+
2
x
)
(
z
+
2
y
)
.
\geq \sqrt{(x+2y)(x+2z)}+\sqrt{(y+2x)(y+2z)}+\sqrt{(z+2x)(z+2y)}.
≥
(
x
+
2
y
)
(
x
+
2
z
)
+
(
y
+
2
x
)
(
y
+
2
z
)
+
(
z
+
2
x
)
(
z
+
2
y
)
.
inequalities