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Korea Junior Mathematics Olympiad
2011 Korea Junior Math Olympiad
7
A discrete inequality looks like continuous one
A discrete inequality looks like continuous one
Source: KJMO 2011 pr 7
October 17, 2017
inequalities
n-variable inequality
convex function
Problem Statement
For those real numbers
x
1
,
x
2
,
…
,
x
2011
x_1 , x_2 , \ldots , x_{2011}
x
1
,
x
2
,
…
,
x
2011
where each of which satisfies
0
≤
x
1
≤
1
0 \le x_1 \le 1
0
≤
x
1
≤
1
(
i
=
1
,
2
,
…
,
2011
i = 1 , 2 , \ldots , 2011
i
=
1
,
2
,
…
,
2011
), find the maximum of
x
1
3
+
x
2
3
+
⋯
+
x
2011
3
−
(
x
1
x
2
x
3
+
x
2
x
3
x
4
+
⋯
+
x
2011
x
1
x
2
)
x_1^3+x_2^3+ \cdots + x_{2011}^3 - \left( x_1x_2x_3 + x_2x_3x_4 + \cdots + x_{2011}x_1x_2 \right)
x
1
3
+
x
2
3
+
⋯
+
x
2011
3
−
(
x
1
x
2
x
3
+
x
2
x
3
x
4
+
⋯
+
x
2011
x
1
x
2
)
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