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2022 Brazil Undergrad MO
5
Product of density-like function is larger than 1/4
Product of density-like function is larger than 1/4
Source: Brazilian Undergrad Mathematics Olympiad 2022 P5
November 26, 2022
Problem Statement
Given
X
⊂
N
X \subset \mathbb{N}
X
⊂
N
, define
d
(
X
)
d(X)
d
(
X
)
as the largest
c
∈
[
0
,
1
]
c \in [0, 1]
c
∈
[
0
,
1
]
such that for any
a
<
c
a < c
a
<
c
and
n
0
∈
N
n_0\in \mathbb{N}
n
0
∈
N
, there exists
m
,
r
∈
N
m, r \in \mathbb{N}
m
,
r
∈
N
with
r
≥
n
0
r \geq n_0
r
≥
n
0
and
∣
X
∩
[
m
,
m
+
r
)
∣
r
≥
a
\frac{\mid X \cap [m, m+r)\mid}{r} \geq a
r
∣
X
∩
[
m
,
m
+
r
)
∣
≥
a
. Let
E
,
F
⊂
N
E, F \subset \mathbb{N}
E
,
F
⊂
N
such that
d
(
E
)
d
(
F
)
>
1
/
4
d(E)d(F) > 1/4
d
(
E
)
d
(
F
)
>
1/4
. Prove that for any prime
p
p
p
and
k
∈
N
k\in\mathbb{N}
k
∈
N
, there exists
m
∈
E
,
n
∈
F
m \in E, n \in F
m
∈
E
,
n
∈
F
such that
m
≡
n
(
m
o
d
p
k
)
m\equiv n \pmod{p^k}
m
≡
n
(
mod
p
k
)
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