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Poland - Second Round
2002 Poland - Second Round
3
Find n for which inequality with x_i,y_i holds
Find n for which inequality with x_i,y_i holds
Source:
February 2, 2011
inequalities
inequalities proposed
Problem Statement
Find all positive integers
n
n
n
such that for all real numbers
x
1
,
x
2
,
…
,
x
n
,
y
1
,
y
2
,
…
,
y
n
x_1,x_2,\ldots ,x_n,y_1,y_2,\ldots ,y_n
x
1
,
x
2
,
…
,
x
n
,
y
1
,
y
2
,
…
,
y
n
the following inequality holds:
x
1
x
2
…
x
n
+
y
1
y
2
…
y
n
≤
x
1
2
+
y
1
2
⋅
x
2
2
+
y
2
2
⋅
⋯
x
n
2
+
y
n
2
⋅
x_1x_2\ldots x_n+y_1y_2\ldots y_n\le\sqrt{x_1^2+y_1^2}\cdot\sqrt{x_2^2+y_2^2}\cdot \cdots \sqrt{x_n^2+y_n^2}\cdot
x
1
x
2
…
x
n
+
y
1
y
2
…
y
n
≤
x
1
2
+
y
1
2
⋅
x
2
2
+
y
2
2
⋅
⋯
x
n
2
+
y
n
2
⋅
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