Question 1202567
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By a general principle, given a set of numbers with a particular sum, the maximum product of the numbers is when the numbers are all equal.<br>
2000 numbers, all equal, with a sum of 2007, means each number is 2007/2000.<br>
The maximum product is<br>
ANSWER: {{{(2007/2000)^2000}}}.<br>
Note the problem is not defined precisely enough.  If some of the numbers are allowed to be negative, then there is no maximum product.  Given a positive number x, the set consisting of -x, -1, 1997 1's, and x+11 has a sum of 2007, and the product of the numbers is x(x+11), which clearly has no maximum.<br>