Problem 5

Part of 2012 CIIM

Problems(1)

CIIM 2012 Problem 5

Source:

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

. A direction function for

D

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

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}

d_j \in D

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

for every positive integer

j

.
Determine the least integer

k

such that for any direction fuction

f

k

compatible reals with

f

then there are infinite reals compatible with

f

CIIMCIIM 2012undergraduatefunction