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German National Olympiad
2016 German National Olympiad
6
Cyclic function in 7 variable with a Maximum
Cyclic function in 7 variable with a Maximum
Source: Germany 2016 - Problem 6
June 19, 2016
algebra
cyclic function
function
Inequality
maximum
Problem Statement
Let
f
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
,
x
6
,
x
7
)
=
x
1
x
2
x
4
+
x
2
x
3
x
5
+
x
3
x
4
x
6
+
x
4
x
5
x
7
+
x
5
x
6
x
1
+
x
6
x
7
x
2
+
x
7
x
1
x
3
f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=x_1x_2x_4+x_2x_3x_5+x_3x_4x_6+x_4x_5x_7+x_5x_6x_1+x_6x_7x_2+x_7x_1x_3
f
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
,
x
6
,
x
7
)
=
x
1
x
2
x
4
+
x
2
x
3
x
5
+
x
3
x
4
x
6
+
x
4
x
5
x
7
+
x
5
x
6
x
1
+
x
6
x
7
x
2
+
x
7
x
1
x
3
be defined for non-negative real numbers
x
1
,
x
2
,
…
,
x
7
x_1,x_2,\dots,x_7
x
1
,
x
2
,
…
,
x
7
with sum
1
1
1
. Prove that
f
(
x
1
,
x
2
,
…
,
x
7
)
f(x_1,x_2,\dots,x_7)
f
(
x
1
,
x
2
,
…
,
x
7
)
has a maximum value and find that value.
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