MathDB
Problems
Contests
National and Regional Contests
Lithuania Contests
Grand Duchy of Lithuania
2023 Grand Duchy of Lithuania
4
4
Part of
2023 Grand Duchy of Lithuania
Problems
(1)
(2^n -1)!! -1 divides 2^n
Source: 2023 Grand Duchy of Lithuania, MC p4 (Baltic Way TST)
3/23/2024
Note that
k
≥
1
k\ge 1
k
≥
1
for an odd natural number
k
!
!
=
k
⋅
(
k
−
2
)
⋅
.
.
.
⋅
1.
k! ! = k \cdot (k - 2) \cdot ... \cdot 1.
k
!!
=
k
⋅
(
k
−
2
)
⋅
...
⋅
1.
Prove that
(
2
n
−
1
)
!
!
−
1
(2^n -1)!! -1
(
2
n
−
1
)!!
−
1
divides
2
n
2^n
2
n
for all
n
≥
3
n \ge 3
n
≥
3
.
number theory
divides
factorial