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Balkan MO Shortlist
2021 Balkan MO Shortlist
N5
BMO Shortlist 2021 N5
BMO Shortlist 2021 N5
Source: BMO Shortlist 2021
May 8, 2022
Balkan
shortlist
2021
number theory
floor function
Problem Statement
A natural number
n
n
n
is given. Determine all
(
n
−
1
)
(n - 1)
(
n
−
1
)
-tuples of nonnegative integers
a
1
,
a
2
,
.
.
.
,
a
n
−
1
a_1, a_2, ..., a_{n - 1}
a
1
,
a
2
,
...
,
a
n
−
1
such that
⌊
m
2
n
−
1
⌋
+
⌊
2
m
+
a
1
2
n
−
1
⌋
+
⌊
2
2
m
+
a
2
2
n
−
1
⌋
+
⌊
2
3
m
+
a
3
2
n
−
1
⌋
+
.
.
.
+
⌊
2
n
−
1
m
+
a
n
−
1
2
n
−
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
⌊
2
n
−
1
m
⌋
+
⌊
2
n
−
1
2
m
+
a
1
⌋
+
⌊
2
n
−
1
2
2
m
+
a
2
⌋
+
⌊
2
n
−
1
2
3
m
+
a
3
⌋
+
...
+
⌊
2
n
−
1
2
n
−
1
m
+
a
n
−
1
⌋
=
m
holds for all
m
∈
Z
m \in \mathbb{Z}
m
∈
Z
.
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