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National and Regional Contests
Austria Contests
Austrian MO National Competition
2004 Federal Competition For Advanced Students, Part 1
3
3
Part of
2004 Federal Competition For Advanced Students, Part 1
Problems
(1)
Z(a,b) is an integer for a \leq b (Austria 2004 -Part1)
Source:
6/18/2011
For natural numbers
a
,
b
a, b
a
,
b
, define
Z
(
a
,
b
)
=
(
3
a
)
!
⋅
(
4
b
)
!
a
!
4
⋅
b
!
3
Z(a,b)=\frac{(3a)!\cdot (4b)!}{a!^4 \cdot b!^3}
Z
(
a
,
b
)
=
a
!
4
⋅
b
!
3
(
3
a
)!
⋅
(
4
b
)!
.(a) Prove that
Z
(
a
,
b
)
Z(a, b)
Z
(
a
,
b
)
is an integer for
a
≤
b
a \leq b
a
≤
b
.(b) Prove that for each natural number
b
b
b
there are infinitely many natural numbers a such that
Z
(
a
,
b
)
Z(a, b)
Z
(
a
,
b
)
is not an integer.
factorial