The sum of the first two numbers and the last two numbers is $1004+1005=2009$. This counts the middle number twice. But the sum of all three numbers is $2009$, so the middle number is $0$. Hence the product of all three numbers is $0$.

[Alternatively: Let the three numbers be a, b and c.

We have

$a+b=1004$

$b+c=1005$

and
$a+b+c=2009$

Adding the first two
equations gives

$a+2b+c=2009$

and subtracting the third
equationg from this gives

$b=0$

Thus the product
$abc=0$.]

*This problem is taken from the UKMT Mathematical Challenges.**View the archive of all weekly problems grouped by curriculum topic*

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