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Find the smallest positive integer that is exactly half the sum of its proper divisors.   

I didn't know what a "proper divisor" is, so I looked it up.  If you use the definition given at wolfram.com, the problem can't be solved.  Wolfram defines proper divisor as follows:  "A positive proper divisor is a positive divisor of a number, excluding itself. For example, 1, 2, and 3 are positive proper divisors of 6, but 6 itself is not." 

6 would work if you didn't have to exclude itself.  Divisors 1 & 2 & 3 & 6 added up equals 12 and 6 is half of that, but Wolfram says 6 cannot be counted as a proper divisor of 6. 

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The "proper divisors" of 120 are as follows:

(1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60) = 240.

Therefore, 120 is  exactly half the sum of its proper divisors.