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National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2018 Macedonia JBMO TST
3
3
Part of
2018 Macedonia JBMO TST
Problems
(1)
2018 JBMO TST- Macedonia, problem 3
Source: 2018 JBMO TST- Macedonia
5/28/2019
Let
x
x
x
,
y
y
y
, and
z
z
z
be positive real numbers such that
x
+
y
+
z
=
1
x + y + z = 1
x
+
y
+
z
=
1
. Prove that
(
x
+
y
)
3
z
+
(
y
+
z
)
3
x
+
(
z
+
x
)
3
y
+
9
x
y
z
≥
9
(
x
y
+
y
z
+
z
x
)
\frac{(x+y)^3}{z} + \frac{(y+z)^3}{x} + \frac{(z+x)^3}{y} + 9xyz \ge 9(xy + yz + zx)
z
(
x
+
y
)
3
+
x
(
y
+
z
)
3
+
y
(
z
+
x
)
3
+
9
x
yz
≥
9
(
x
y
+
yz
+
z
x
)
.When does equality hold?
JMMO
2018
Macedonia
algebra
Inequality
inequalities