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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2012 Korea Junior Math Olympiad
7
7
Part of
2012 Korea Junior Math Olympiad
Problems
(1)
max of \frac{(\sqrt{s_1x_1} +...+\sqrt{s_5x_5})^2}{a_1x_1+...+a_5x_5}
Source: KJMO 2012 p7
5/4/2019
If all
x
k
x_k
x
k
(
k
=
1
,
2
,
3
,
4
,
5
)
k = 1, 2, 3, 4, 5)
k
=
1
,
2
,
3
,
4
,
5
)
are positive reals, and
{
a
1
,
a
2
,
a
3
,
a
4
,
a
5
}
=
{
1
,
2
,
3
,
4
,
5
}
\{a_1,a_2, a_3, a_4, a_5\} = \{1, 2,3 , 4, 5\}
{
a
1
,
a
2
,
a
3
,
a
4
,
a
5
}
=
{
1
,
2
,
3
,
4
,
5
}
, find the maximum of
(
s
1
x
1
+
s
2
x
2
+
s
3
x
3
+
s
4
x
4
+
s
5
x
5
)
2
a
1
x
1
+
a
2
x
2
+
a
3
x
3
+
a
4
x
4
+
a
5
x
5
\frac{(\sqrt{s_1x_1} +\sqrt{s_2x_2}+\sqrt{s_3x_3}+\sqrt{s_4x_4}+\sqrt{s_5x_5})^2}{a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 + a_5x_5}
a
1
x
1
+
a
2
x
2
+
a
3
x
3
+
a
4
x
4
+
a
5
x
5
(
s
1
x
1
+
s
2
x
2
+
s
3
x
3
+
s
4
x
4
+
s
5
x
5
)
2
(
s
k
=
a
1
+
a
2
+
.
.
.
+
a
k
s_k = a_1 + a_2 +... + a_k
s
k
=
a
1
+
a
2
+
...
+
a
k
)
maximum
positive real
Sum
algebra
inequalities