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A set with prod

Source: Indian RMO 2003 Problem 7

October 27, 2005

factorialratio

Problem Statement

Consider the set

X

=

\{ 1,2 \ldots 10 \}

. Find two disjoint nonempty sunsets

A

and

B

of

X

such that a)

A \cup B = X

; b)

\prod_{x\in A}x

is divisible by

\prod_{x\in B}x

, where

\prod_{x\in C}x

is the product of all numbers in

C

; c)

\frac{ \prod\limits_{x\in A}x}{ \prod\limits_{x\in B}x}

is as small as possible.

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