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International Zhautykov Olympiad
2022 International Zhautykov Olympiad
6
6
Part of
2022 International Zhautykov Olympiad
Problems
(1)
Bounded Sequences
Source: IZhO 2022 Day 2 Problem 6
2/18/2022
Do there exist two bounded sequences
a
1
,
a
2
,
…
a_1, a_2,\ldots
a
1
,
a
2
,
…
and
b
1
,
b
2
,
…
b_1, b_2,\ldots
b
1
,
b
2
,
…
such that for each positive integers
n
n
n
and
m
>
n
m>n
m
>
n
at least one of the two inequalities
∣
a
m
−
a
n
∣
>
1
/
n
,
|a_m-a_n|>1/\sqrt{n},
∣
a
m
−
a
n
∣
>
1/
n
,
and
∣
b
m
−
b
n
∣
>
1
/
n
|b_m-b_n|>1/\sqrt{n}
∣
b
m
−
b
n
∣
>
1/
n
holds?
algebra
Sequence
combinatorics
izho