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Problems
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International Contests
Austrian-Polish
1985 Austrian-Polish Competition
7
7
Part of
1985 Austrian-Polish Competition
Problems
(1)
Maximize 4-variable ratio
Source: Austrian-Polish 1985, Problem 7
7/5/2015
Find an upper bound for the ratio
x
1
x
2
+
2
x
2
x
3
+
x
3
x
4
x
1
2
+
x
2
2
+
x
3
2
+
x
4
2
\frac{x_1x_2+2x_2x_3+x_3x_4}{x_1^2+x_2^2+x_3^2+x_4^2}
x
1
2
+
x
2
2
+
x
3
2
+
x
4
2
x
1
x
2
+
2
x
2
x
3
+
x
3
x
4
over all quadruples of real numbers
(
x
1
,
x
2
,
x
3
,
x
4
)
≠
(
0
,
0
,
0
,
0
)
(x_1,x_2,x_3,x_4)\neq (0,0,0,0)
(
x
1
,
x
2
,
x
3
,
x
4
)
=
(
0
,
0
,
0
,
0
)
.Note. The smaller the bound, the better the solution.
inequalities
maximization
ratio
four variables