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In Sum-mary

Stage: 3 Short Challenge Level: Challenge Level:1

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.
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