2013 USAJMO Problems/Problem 4
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Problem
Let be the number of ways to write
as a sum of powers of
, where we keep track of the order of the summation. For example,
because
can be written as
,
,
,
,
, and
. Find the smallest
greater than
for which
is odd.
Solution
First of all, note that =
where
is the largest integer such that
. We let
for convenience.
From here, we proceed by induction, with our claim being that the only such that
is odd are
representable of the form