2014 USAJMO Problems/Problem 4
Problem
Let be an integer, and let
denote the sum of the digits of
when it is written in base
. Show that there are infinitely many positive integers that cannot be represented in the form
, where
is a positive integer.
Solution
Define , and call a number unrepresentable if it cannot equal
for a positive integer
.
We claim that in the interval
there exists an unrepresentable number, for every positive integer
.
If is unrepresentable, we're done. Otherwise, time for our lemma:
Lemma: Define the function to equal the number of integer x less than
such that
. If
for some x, then
.
Proof: Let be the set of integers x less than
such that
. Then for every integer in
, append the digit
to the front of it to create a valid integer in
. Also, notice that
. Removing the digit
from the front of x creates a number that is not in
. Hence, $F(p) \arrow F(p+1)$ (Error compiling LaTeX. Unknown error_msg), but there exists an element of
not corresponding with
, so
.
Note that our lemma combined with the Pigeonhole Principle essentially proves the claim. Therefore, because there are infinitely many intervals containing an unrepresentable number, there are infinitely many unrepresentable numbers.