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2022 Azerbaijan Junior National Olympiad
A3
Cyclic inequality where x^2+y^2+z^2=x+y+z
Cyclic inequality where x^2+y^2+z^2=x+y+z
Source: Azerbaijan 2022 Junior National Olympiad
May 14, 2022
Inequality
algebra
Azerbaijan
Junior
inequalities
Problem Statement
Let
x
,
y
,
z
∈
R
+
x,y,z \in \mathbb{R}^{+}
x
,
y
,
z
∈
R
+
and
x
2
+
y
2
+
z
2
=
x
+
y
+
z
x^2+y^2+z^2=x+y+z
x
2
+
y
2
+
z
2
=
x
+
y
+
z
. Prove that
x
+
y
+
z
+
3
≥
6
x
y
+
y
z
+
z
x
3
3
x+y+z+3 \ge 6 \sqrt[3]{\frac{xy+yz+zx}{3}}
x
+
y
+
z
+
3
≥
6
3
3
x
y
+
yz
+
z
x
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