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Omni-MATH / A natural number n is given. Determine all (n−1)-tuples of nonnegative integers a₁,a₂,...,a_(n−1) such that …

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problem

A natural number nn is given. Determine all (n−1)(n - 1)-tuples of nonnegative integers a1,a2,...,an−1a_1, a_2, ..., a_{n - 1} such that ⌊m2n−1⌋+⌊2m+a12n−1⌋+⌊22m+a22n−1⌋+⌊23m+a32n−1⌋+...+⌊2n−1m+an−12n−1⌋=m\lfloor \frac{m}{2^n - 1}\rfloor + \lfloor \frac{2m + a_1}{2^n - 1}\rfloor + \lfloor \frac{2^2m + a_2}{2^n - 1}\rfloor + \lfloor \frac{2^3m + a_3}{2^n - 1}\rfloor + ... + \lfloor \frac{2^{n - 1}m + a_{n - 1}}{2^n - 1}\rfloor = m holds for all m∈Zm \in \mathbb{Z}.
Plain-text mathematical notation (without MathML)
A natural number n is given. Determine all (n−1)-tuples of nonnegative integers a₁,a₂,...,a_(n−1) such that
⌊(m)/(2^(n)−1)⌋+⌊(2m+a₁)/(2^(n)−1)⌋+⌊(2²m+a₂)/(2^(n)−1)⌋+⌊(2³m+a₃)/(2^(n)−1)⌋+...+⌊(2^(n−1)m+a_(n−1))/(2^(n)−1)⌋=m
holds for all m∈Z.
Original LaTeX notation
A natural number $n$ is given. Determine all $(n - 1)$-tuples of nonnegative integers $a_1, a_2, ..., a_{n - 1}$ such that
$$\lfloor \frac{m}{2^n - 1}\rfloor + \lfloor \frac{2m + a_1}{2^n - 1}\rfloor + \lfloor \frac{2^2m + a_2}{2^n - 1}\rfloor + \lfloor \frac{2^3m + a_3}{2^n - 1}\rfloor + ... + \lfloor \frac{2^{n - 1}m + a_{n - 1}}{2^n - 1}\rfloor = m$$
holds for all $m \in \mathbb{Z}$.

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Source and history

task_id
a4c0eaf0-ab9b-5fce-838d-6fd89fbc56cb
task_key
test--a4c0eaf0-ab9b-5fce-838d-6fd89fbc56cb
task_revision_id
1
config / split
/ test

Official source

initial import