MathDB

N5

Part of 2021 Balkan MO Shortlist

Problems(1)

BMO Shortlist 2021 N5

Source: BMO Shortlist 2021

5/8/2022
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}.
Balkanshortlist2021number theoryfloor function