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2
Algebra Inequality
Algebra Inequality
Source: Archimedes Junior 2008
March 17, 2020
inequalities
Problem Statement
If
x
,
y
,
z
x,y,z
x
,
y
,
z
are positive real numbers with
x
2
+
y
2
+
z
2
=
3
x^2+y^2+z^2=3
x
2
+
y
2
+
z
2
=
3
, prove that
3
2
<
1
+
y
2
x
+
2
+
1
+
z
2
y
+
2
+
1
+
x
2
z
+
2
<
3
\frac32<\frac{1+y^2}{x+2}+\frac{1+z^2}{y+2}+\frac{1+x^2}{z+2}<3
2
3
<
x
+
2
1
+
y
2
+
y
+
2
1
+
z
2
+
z
+
2
1
+
x
2
<
3
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