For an integer
, denote by
the unique number in
such that
is a multiple of
. A function
satisfies
,
,
and
for all integers
,
with
. Prove that
holds for all integers
.
Proposed by Gerhard Woeginger, Austria
%V0
For an integer $m$, denote by $t(m)$ the unique number in $\{1, 2, 3\}$ such that $m + t(m)$ is a multiple of $3$. A function $f: \mathbb{Z}\to\mathbb{Z}$ satisfies $f( - 1) = 0$, $f(0) = 1$, $f(1) = - 1$ and $f\left(2^{n} + m\right) = f\left(2^n - t(m)\right) - f(m)$ for all integers $m$, $n\ge 0$ with $2^n > m$. Prove that $f(3p)\ge 0$ holds for all integers $p\ge 0$.
Proposed by Gerhard Woeginger, Austria
Školjka