MathDB
Problems
Contests
International Contests
Tournament Of Towns
1986 Tournament Of Towns
(129) 4
TOT 129 1986 Autumn S4 1987 /( 1 986 !! + 1985 !! )
TOT 129 1986 Autumn S4 1987 /( 1 986 !! + 1985 !! )
Source:
August 29, 2019
factorial
number theory
Divide
divisible
divisor
Problem Statement
We define
N
!
!
N !!
N
!!
to be
N
(
N
−
2
)
(
N
−
4
)
.
.
.
5
⋅
3
⋅
1
N(N - 2)(N -4)...5 \cdot 3 \cdot 1
N
(
N
−
2
)
(
N
−
4
)
...5
⋅
3
⋅
1
if
N
N
N
is odd and
N
(
N
−
2
)
(
N
−
4
)
.
.
.
6
⋅
4
⋅
2
N(N -2)(N -4)... 6\cdot 4\cdot 2
N
(
N
−
2
)
(
N
−
4
)
...6
⋅
4
⋅
2
if
N
N
N
is even . For example,
8
!
!
=
8
⋅
6
⋅
4
⋅
2
8 !! = 8 \cdot 6\cdot 4\cdot 2
8
!!
=
8
⋅
6
⋅
4
⋅
2
, and
9
!
!
=
9
v
7
⋅
5
⋅
3
⋅
1
9 !! = 9v 7 \cdot 5\cdot 3 \cdot 1
9
!!
=
9
v
7
⋅
5
⋅
3
⋅
1
. Prove that
1986
!
!
+
1985
!
!
1986 !! + 1985 !!
1986
!!
+
1985
!!
i s divisible by
1987
1987
1987
.(V.V . Proizvolov , Moscow)
Back to Problems
View on AoPS