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2022 JBMO Shortlist
A2
JBMO Shortlist 2022 A2
JBMO Shortlist 2022 A2
Source: JBMO Shortlist 2022
June 26, 2023
Inequality
Junior
Balkan
shortlist
algebra
Problem Statement
Let
x
,
y
,
x, y,
x
,
y
,
and
z
z
z
be positive real numbers such that
x
y
+
y
z
+
z
x
=
3
xy + yz + zx = 3
x
y
+
yz
+
z
x
=
3
. Prove that
x
+
3
y
+
z
+
y
+
3
z
+
x
+
z
+
3
x
+
y
+
3
≥
27
⋅
(
x
+
y
+
z
)
2
(
x
+
y
+
z
)
3
.
\frac{x + 3}{y + z} + \frac{y + 3}{z + x} + \frac{z + 3}{x + y} + 3 \ge 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^2}{(x + y + z)^3}.
y
+
z
x
+
3
+
z
+
x
y
+
3
+
x
+
y
z
+
3
+
3
≥
27
⋅
(
x
+
y
+
z
)
3
(
x
+
y
+
z
)
2
.
Proposed by Petar Filipovski, Macedonia
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