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Hungary-Israel Binational
2003 Hungary-Israel Binational
1
n positive reals
n positive reals
Source: 16-th Hungary-Israel Binational Mathematical Competition 2003
March 30, 2007
inequalities
inequalities proposed
Problem Statement
If
x
1
,
x
2
,
.
.
.
,
x
n
x_{1}, x_{2}, . . . , x_{n}
x
1
,
x
2
,
...
,
x
n
are positive numbers, prove the inequality
x
1
3
x
1
2
+
x
1
x
2
+
x
2
2
+
x
2
3
x
2
2
+
x
2
x
3
+
x
3
2
+
.
.
.
+
x
n
3
x
n
2
+
x
n
x
1
+
x
1
2
≥
x
1
+
x
2
+
.
.
.
+
x
n
3
\frac{x_{1}^{3}}{x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}}+\frac{x_{2}^{3}}{x_{2}^{2}+x_{2}x_{3}+x_{3}^{2}}+...+\frac{x_{n}^{3}}{x_{n}^{2}+x_{n}x_{1}+x_{1}^{2}}\geq\frac{x_{1}+x_{2}+...+x_{n}}{3}
x
1
2
+
x
1
x
2
+
x
2
2
x
1
3
+
x
2
2
+
x
2
x
3
+
x
3
2
x
2
3
+
...
+
x
n
2
+
x
n
x
1
+
x
1
2
x
n
3
≥
3
x
1
+
x
2
+
...
+
x
n
.
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