MathDB
Problems
Contests
National and Regional Contests
PEN Problems
PEN G Problems
22
22
Part of
PEN G Problems
Problems
(1)
G 22
Source:
5/25/2007
For a positive real number
α
\alpha
α
, define
S
(
α
)
=
{
⌊
n
α
⌋
∣
n
=
1
,
2
,
3
,
⋯
}
.
S(\alpha)=\{ \lfloor n\alpha\rfloor \; \vert \; n=1,2,3,\cdots \}.
S
(
α
)
=
{⌊
n
α
⌋
∣
n
=
1
,
2
,
3
,
⋯
}
.
Prove that
N
\mathbb{N}
N
cannot be expressed as the disjoint union of three sets
S
(
α
)
S(\alpha)
S
(
α
)
,
S
(
β
)
S(\beta)
S
(
β
)
, and
S
(
γ
)
S(\gamma)
S
(
γ
)
.
floor function
Irrational numbers