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National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1967 Dutch Mathematical Olympiad
5
sequence ([n x])
sequence ([n x])
Source: Netherlands - Dutch NMO 1967 p5
January 31, 2023
number theory
floor function
algebra
Problem Statement
Consider rows of the form:
[
x
]
,
[
2
x
]
,
[
3
x
]
,
.
.
.
[x], [2x], [3x], ...
[
x
]
,
[
2
x
]
,
[
3
x
]
,
...
Proof that, if
N
∈
N
N \in N
N
∈
N
does not occur in the sequence
(
[
n
x
]
)
([n x])
([
n
x
])
, then there is an
n
∈
N
n \in N
n
∈
N
with
n
−
1
<
N
x
<
n
−
1
x
n - 1 < \frac{N}{x}< n -\frac{1}{x}
n
−
1
<
x
N
<
n
−
x
1
Prove that, for
x
,
y
∉
Q
x, y \notin Q
x
,
y
∈
/
Q
:
1
x
+
1
y
=
1
\frac{1}{x}+\frac{1}{y} = 1
x
1
+
y
1
=
1
, then each
N
∈
N
N \in N
N
∈
N
term is either of
(
[
n
x
]
)
([nx])
([
n
x
])
or of
(
[
n
y
]
)
([ny])
([
n
y
])
.
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