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National and Regional Contests
Turkey Contests
Turkey EGMO TST
2014 Turkey EGMO TST
4
Inequality (Turkey EGMO TST 2014)
Inequality (Turkey EGMO TST 2014)
Source: (Turkey EGMO TST 2014)
July 28, 2016
inequalities
Problem Statement
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive real numbers such that
x
+
y
+
z
≥
x
2
+
y
2
+
z
2
x+y+z \ge x^2+y^2+z^2
x
+
y
+
z
≥
x
2
+
y
2
+
z
2
. Show that;
x
2
+
3
x
3
+
1
+
y
2
+
3
y
3
+
1
+
z
2
+
3
z
3
+
1
≥
6
\dfrac{x^2+3}{x^3+1}+\dfrac{y^2+3}{y^3+1}+\dfrac{z^2+3}{z^3+1}\ge6
x
3
+
1
x
2
+
3
+
y
3
+
1
y
2
+
3
+
z
3
+
1
z
2
+
3
≥
6
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