MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2023 China Team Selection Test
P14
P14
Part of
2023 China Team Selection Test
Problems
(1)
Finally an inequality
Source: China TST 2023 Problem 14
3/27/2023
For any nonempty, finite set
B
B
B
and real
x
x
x
, define
d
B
(
x
)
=
min
b
∈
B
∣
x
−
b
∣
d_B(x) = \min_{b\in B} |x-b|
d
B
(
x
)
=
b
∈
B
min
∣
x
−
b
∣
(1) Given positive integer
m
m
m
. Find the smallest real number
λ
\lambda
λ
(possibly depending on
m
m
m
) such that for any positive integer
n
n
n
and any reals
x
1
,
⋯
,
x
n
∈
[
0
,
1
]
x_1,\cdots,x_n \in [0,1]
x
1
,
⋯
,
x
n
∈
[
0
,
1
]
, there exists an
m
m
m
-element set
B
B
B
of real numbers satisfying
d
B
(
x
1
)
+
⋯
+
d
B
(
x
n
)
≤
λ
n
d_B(x_1)+\cdots+d_B(x_n) \le \lambda n
d
B
(
x
1
)
+
⋯
+
d
B
(
x
n
)
≤
λn
(2) Given positive integer
m
m
m
and positive real
ϵ
\epsilon
ϵ
. Prove that there exists a positive integer
n
n
n
and nonnegative reals
x
1
,
⋯
,
x
n
x_1,\cdots,x_n
x
1
,
⋯
,
x
n
, satisfying for any
m
m
m
-element set
B
B
B
of real numbers, we have
d
B
(
x
1
)
+
⋯
+
d
B
(
x
n
)
>
(
1
−
ϵ
)
(
x
1
+
⋯
+
x
n
)
d_B(x_1)+\cdots+d_B(x_n) > (1-\epsilon)(x_1+\cdots+x_n)
d
B
(
x
1
)
+
⋯
+
d
B
(
x
n
)
>
(
1
−
ϵ
)
(
x
1
+
⋯
+
x
n
)
inequalities
Sets