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2021-IMOC
A10
A hard ineq, given that xyz + xy + yz + zx = 4
A hard ineq, given that xyz + xy + yz + zx = 4
Source: IMOC 2021 A10
August 11, 2021
Inequality
algebra
inequalities
Problem Statement
For any positive reals
x
x
x
,
y
y
y
,
z
z
z
with
x
y
z
+
x
y
+
y
z
+
z
x
=
4
xyz + xy + yz + zx = 4
x
yz
+
x
y
+
yz
+
z
x
=
4
, prove that
x
y
+
x
+
y
z
+
y
z
+
y
+
z
x
+
z
x
+
z
+
x
y
≥
3
3
(
x
+
2
)
(
y
+
2
)
(
z
+
2
)
(
2
x
+
1
)
(
2
y
+
1
)
(
2
z
+
1
)
.
\sqrt{\frac{xy+x+y}{z}}+\sqrt{\frac{yz+y+z}{x}}+\sqrt{\frac{zx+z+x}{y}}\geq 3\sqrt{\frac{3(x+2)(y+2)(z+2)}{(2x + 1)(2y + 1)(2z + 1). }}
z
x
y
+
x
+
y
+
x
yz
+
y
+
z
+
y
z
x
+
z
+
x
≥
3
(
2
x
+
1
)
(
2
y
+
1
)
(
2
z
+
1
)
.
3
(
x
+
2
)
(
y
+
2
)
(
z
+
2
)
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