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2010 Danube Mathematical Olympiad
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5
Part of
2010 Danube Mathematical Olympiad
Problems
(1)
please help!!!! problem from danube cup romania 2010
Source: Donova Mathmatical Olympiad 2010
2/5/2011
Let
n
≥
3
n\ge3
n
≥
3
be a positive integer. Find the real numbers
x
1
≥
0
,
…
,
x
n
≥
0
x_1\ge0,\ldots,x_n\ge 0
x
1
≥
0
,
…
,
x
n
≥
0
, with
x
1
+
x
2
+
…
+
x
n
=
n
x_1+x_2+\ldots +x_n=n
x
1
+
x
2
+
…
+
x
n
=
n
, for which the expression
(
n
−
1
)
(
x
1
2
+
x
2
2
+
…
+
x
n
2
)
+
n
x
1
x
2
…
x
n
(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n
(
n
−
1
)
(
x
1
2
+
x
2
2
+
…
+
x
n
2
)
+
n
x
1
x
2
…
x
n
takes a minimal value.
algebra
polynomial
function
inequalities
calculus
n-variable inequality