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China National Olympiad
2012 China National Olympiad
1
Find the maximum of F
Find the maximum of F
Source: 2012 China Mathematical Olympaid P4
January 13, 2012
inequalities
inequalities proposed
Problem Statement
Let
f
(
x
)
=
(
x
+
a
)
(
x
+
b
)
f(x)=(x + a)(x + b)
f
(
x
)
=
(
x
+
a
)
(
x
+
b
)
where
a
,
b
>
0
a,b>0
a
,
b
>
0
. For any reals
x
1
,
x
2
,
…
,
x
n
⩾
0
x_1,x_2,\ldots ,x_n\geqslant 0
x
1
,
x
2
,
…
,
x
n
⩾
0
satisfying
x
1
+
x
2
+
…
+
x
n
=
1
x_1+x_2+\ldots +x_n =1
x
1
+
x
2
+
…
+
x
n
=
1
, find the maximum of
F
=
∑
1
⩽
i
<
j
⩽
n
min
{
f
(
x
i
)
,
f
(
x
j
)
}
F=\sum\limits_{1 \leqslant i < j \leqslant n} {\min \left\{ {f({x_i}),f({x_j})} \right\}}
F
=
1
⩽
i
<
j
⩽
n
∑
min
{
f
(
x
i
)
,
f
(
x
j
)
}
.
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