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National and Regional Contests
China Contests
China Team Selection Test
2023 China Team Selection Test
P10
P10
Part of
2023 China Team Selection Test
Problems
(1)
combinatorics problem
Source: 2023 China TST P10
3/19/2023
The set of nonempty integers
A
A
A
is said to be "elegant" if it is for any
a
∈
A
,
a\in A,
a
∈
A
,
1
≤
k
≤
2023
,
1\leq k\leq 2023,
1
≤
k
≤
2023
,
∣
{
b
∈
A
:
⌊
b
3
k
⌋
=
⌊
a
3
k
⌋
}
∣
=
2
k
.
\left| \left\{ b\in A:\left\lfloor\frac b{3^k}\right\rfloor =\left\lfloor\frac a{3^k}\right\rfloor\right\}\right| =2^k.
{
b
∈
A
:
⌊
3
k
b
⌋
=
⌊
3
k
a
⌋
}
=
2
k
.
Prove that if the intersection of the integer set
S
S
S
and any "elegant" set is not empty
,
,
,
then
S
S
S
contains an "elegant" set.
combinatorics
China TST