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Putnam
1995 Putnam
6
Putnam 1995 B6
Putnam 1995 B6
Source: Putnam
January 10, 2007
Putnam
floor function
ceiling function
limit
college contests
Problem Statement
For any
a
>
0
a>0
a
>
0
,set
S
(
a
)
=
{
⌊
n
a
⌋
∣
n
∈
N
}
\mathcal{S}(a)=\{\lfloor{na}\rfloor|n\in \mathbb{N}\}
S
(
a
)
=
{⌊
na
⌋
∣
n
∈
N
}
. Show that there are no three positive reals
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
S
(
a
)
∩
S
(
b
)
=
S
(
b
)
∩
S
(
c
)
=
S
(
c
)
∩
S
(
a
)
=
∅
\mathcal{S}(a)\cap \mathcal{S}(b)=\mathcal{S}(b)\cap \mathcal{S}(c)=\mathcal{S}(c)\cap \mathcal{S}(a)=\emptyset
S
(
a
)
∩
S
(
b
)
=
S
(
b
)
∩
S
(
c
)
=
S
(
c
)
∩
S
(
a
)
=
∅
S
(
a
)
∪
S
(
b
)
∪
S
(
c
)
=
N
\mathcal{S}(a)\cup \mathcal{S}(b)\cup \mathcal{S}(c)=\mathbb{N}
S
(
a
)
∪
S
(
b
)
∪
S
(
c
)
=
N
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