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National High School Mathematics League
2003 National High School Mathematics League
12
Limitation
Limitation
Source: 2003 National High School Mathematics League, Exam One, Problem 12
March 16, 2020
Problem Statement
M
n
=
{
0.
a
1
a
2
⋯
a
n
‾
∣
a
i
∈
0
,
1
,
i
=
1
,
2
,
⋯
,
n
,
a
n
=
1
}
M_n=\{\overline{0.a_1a_2\cdots a_n}|a_i\in{0,1},i=1,2,\cdots,n,a_n=1\}
M
n
=
{
0.
a
1
a
2
⋯
a
n
∣
a
i
∈
0
,
1
,
i
=
1
,
2
,
⋯
,
n
,
a
n
=
1
}
.
T
n
=
∣
M
n
∣
,
S
n
=
∑
x
∈
M
n
x
T_n=|M_n|,S_n=\sum_{x\in M_n}x
T
n
=
∣
M
n
∣
,
S
n
=
∑
x
∈
M
n
x
, then
lim
n
→
∞
S
n
T
n
=
\lim_{n\to\infty}\frac{S_n}{T_n}=
lim
n
→
∞
T
n
S
n
=
________.
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