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2009 Tournament Of Towns
4
New factorial function - TT 2009 Senior-A4
New factorial function - TT 2009 Senior-A4
Source:
September 3, 2010
factorial
function
floor function
number theory proposed
number theory
Problem Statement
Denote by
[
n
]
!
[n]!
[
n
]!
the product
1
⋅
11
⋅
111
⋅
.
.
.
⋅
111...1
⏟
n ones
1 \cdot 11 \cdot 111\cdot ... \cdot \underbrace{111...1}_{\text{n ones}}
1
⋅
11
⋅
111
⋅
...
⋅
n ones
111...1
.(
n
n
n
factors in total). Prove that
[
n
+
m
]
!
[n + m]!
[
n
+
m
]!
is divisible by
[
n
]
!
×
[
m
]
!
[n]! \times [m]!
[
n
]!
×
[
m
]!
(8 points)
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