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National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2024 Junior Balkan Team Selection Tests - Moldova
1
a+b+c=xyz=1 inequality in 6 variables
a+b+c=xyz=1 inequality in 6 variables
Source: Moldova JTST 2024 P1
June 10, 2024
inequalities
Problem Statement
Let
a
,
b
,
c
,
x
,
y
,
z
a,b,c,x,y,z
a
,
b
,
c
,
x
,
y
,
z
be positive real numbers, such that
a
+
b
+
c
=
x
y
z
=
1
a+b+c=xyz=1
a
+
b
+
c
=
x
yz
=
1
Prove that:
x
2
3
a
+
2
+
y
2
3
b
+
2
+
z
2
3
c
+
2
≥
1
\frac{x^2}{3a+2}+\frac{y^2}{3b+2}+\frac{z^2}{3c+2} \ge 1
3
a
+
2
x
2
+
3
b
+
2
y
2
+
3
c
+
2
z
2
≥
1
When does equality hold?
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