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IMO Longlists
1987 IMO Longlists
64
64
Part of
1987 IMO Longlists
Problems
(1)
IMO LongList 1987 - Base-r expansion
Source:
9/6/2010
Let
r
>
1
r > 1
r
>
1
be a real number, and let
n
n
n
be the largest integer smaller than
r
r
r
. Consider an arbitrary real number
x
x
x
with
0
≤
x
≤
n
r
−
1
.
0 \leq x \leq \frac{n}{r-1}.
0
≤
x
≤
r
−
1
n
.
By a base-
r
r
r
expansion of
x
x
x
we mean a representation of
x
x
x
in the form
x
=
a
1
r
+
a
2
r
2
+
a
3
r
3
+
⋯
x=\frac{a_1}{r} + \frac{a_2}{r^2}+\frac{a_3}{r^3}+\cdots
x
=
r
a
1
+
r
2
a
2
+
r
3
a
3
+
⋯
where the
a
i
a_i
a
i
are integers with
0
≤
a
i
<
r
.
0 \leq a_i < r.
0
≤
a
i
<
r
.
You may assume without proof that every number
x
x
x
with
0
≤
x
≤
n
r
−
1
0 \leq x \leq \frac{n}{r-1}
0
≤
x
≤
r
−
1
n
has at least one base-
r
r
r
expansion.Prove that if
r
r
r
is not an integer, then there exists a number
p
p
p
,
0
≤
p
≤
n
r
−
1
0 \leq p \leq \frac{n}{r-1}
0
≤
p
≤
r
−
1
n
, which has infinitely many distinct base-
r
r
r
expansions.
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