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Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2003 India Regional Mathematical Olympiad
7
7
Part of
2003 India Regional Mathematical Olympiad
Problems
(1)
A set with prod
Source: Indian RMO 2003 Problem 7
10/27/2005
Consider the set
X
X
X
=
{
1
,
2
…
10
}
\{ 1,2 \ldots 10 \}
{
1
,
2
…
10
}
. Find two disjoint nonempty sunsets
A
A
A
and
B
B
B
of
X
X
X
such that a)
A
∪
B
=
X
A \cup B = X
A
∪
B
=
X
; b)
∏
x
∈
A
x
\prod_{x\in A}x
∏
x
∈
A
x
is divisible by
∏
x
∈
B
x
\prod_{x\in B}x
∏
x
∈
B
x
, where
∏
x
∈
C
x
\prod_{x\in C}x
∏
x
∈
C
x
is the product of all numbers in
C
C
C
; c)
∏
x
∈
A
x
∏
x
∈
B
x
\frac{ \prod\limits_{x\in A}x}{ \prod\limits_{x\in B}x}
x
∈
B
∏
x
x
∈
A
∏
x
is as small as possible.
factorial
ratio