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2011 Singapore MO Open
3
1/x+1/y+1/z<1/xyz
1/x+1/y+1/z<1/xyz
Source: Singapore MO 2011 open round 2 Q3
July 2, 2011
inequalities
trigonometry
Problem Statement
Let
x
,
y
,
z
>
0
x,y,z>0
x
,
y
,
z
>
0
such that
1
x
+
1
y
+
1
z
<
1
x
y
z
\frac1x+\frac1y+\frac1z<\frac{1}{xyz}
x
1
+
y
1
+
z
1
<
x
yz
1
. Show that
2
x
1
+
x
2
+
2
y
1
+
y
2
+
2
z
1
+
z
2
<
3.
\frac{2x}{\sqrt{1+x^2}}+\frac{2y}{\sqrt{1+y^2}}+\frac{2z}{\sqrt{1+z^2}}<3.
1
+
x
2
2
x
+
1
+
y
2
2
y
+
1
+
z
2
2
z
<
3.
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