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IMOC Shortlist
2021-IMOC
A3
A inequality
A inequality
Source: IMOC 2021 A3
August 11, 2021
inequalities
algebra
Problem Statement
For any real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
with
x
y
z
+
x
+
y
+
z
=
4
,
xyz + x + y + z = 4,
x
yz
+
x
+
y
+
z
=
4
,
show that
(
y
z
+
6
)
2
+
(
z
x
+
6
)
2
+
(
x
y
+
6
)
2
≥
8
(
x
y
z
+
5
)
.
(yz + 6)^2 + (zx + 6)^2 + (xy + 6)^2 \geq 8 (xyz + 5).
(
yz
+
6
)
2
+
(
z
x
+
6
)
2
+
(
x
y
+
6
)
2
≥
8
(
x
yz
+
5
)
.
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