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National and Regional Contests
China Contests
(China) National High School Mathematics League
2022 China Second Round A2
1
Inequality problem with max and min
Inequality problem with max and min
Source: 2022 China Second Round A2
December 22, 2022
inequalities
Problem Statement
a
1
,
a
2
,
.
.
.
,
a
9
a_1,a_2,...,a_9
a
1
,
a
2
,
...
,
a
9
are nonnegative reals with sum
1
1
1
. Define
S
S
S
and
T
T
T
as below:
S
=
min
{
a
1
,
a
2
}
+
2
min
{
a
2
,
a
3
}
+
.
.
.
+
9
min
{
a
9
,
a
1
}
S=\min\{a_1,a_2\}+2\min\{a_2,a_3\}+...+9\min\{a_9,a_1\}
S
=
min
{
a
1
,
a
2
}
+
2
min
{
a
2
,
a
3
}
+
...
+
9
min
{
a
9
,
a
1
}
T
=
max
{
a
1
,
a
2
}
+
2
max
{
a
2
,
a
3
}
+
.
.
.
+
9
max
{
a
9
,
a
1
}
T=\max\{a_1,a_2\}+2\max\{a_2,a_3\}+...+9\max\{a_9,a_1\}
T
=
max
{
a
1
,
a
2
}
+
2
max
{
a
2
,
a
3
}
+
...
+
9
max
{
a
9
,
a
1
}
When
S
S
S
reaches its maximum, find all possible values of
T
T
T
.
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