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CIIM 2012 Problem 5

Source:

August 8, 2016

CIIMCIIM 2012undergraduatefunction

Problem Statement

Let

D=\{0,1,\dots,9\}

. A direction function for

D

is a function

f:D \times D \to \{0,1\}.

A real

r\in [0,1]

is compatible with

f

if it can be written in the form

r = \sum_{j=1}^{\infty} \frac{d_j}{10^j}

with

d_j \in D

and

f(d_j,d_{j+1})=1

for every positive integer

j

.
Determine the least integer

k

such that for any direction fuction

f

, if there are

k

compatible reals with

f

then there are infinite reals compatible with

f

.

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