MathDB

IMO LongList 1987 - Base-r expansion

Source:

September 6, 2010

algebra proposedalgebra

Problem Statement

Let

r > 1

be a real number, and let

n

be the largest integer smaller than

r

. Consider an arbitrary real number

x

with

0 \leq x \leq \frac{n}{r-1}.

By a base-

r

expansion of

x

we mean a representation of

x

in the form

x=\frac{a_1}{r} + \frac{a_2}{r^2}+\frac{a_3}{r^3}+\cdots

where the

a_i

are integers with

0 \leq a_i < r.

You may assume without proof that every number

x

with

0 \leq x \leq \frac{n}{r-1}

has at least one base-

r

expansion.
Prove that if

r

is not an integer, then there exists a number

p

,

0 \leq p \leq \frac{n}{r-1}

, which has infinitely many distinct base-

r

expansions.

Back to Problems View on AoPS