2021 Fall AMC 12B Problems/Problem 5
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Problem 5
Call a fraction 
, not necessarily in the simplest form, special if 
and 
are positive integers whose sum is 
. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
Solution
so the fraction is 
which is 
. We can just ignore the 
part and only care about 
. Now we just group 
as the integers and 
as the halves. We get 
from the integers group and 
from the halves group (easily calculated by halving the values from the integers group). These are both 
integers and we see that 
overlaps, so the answer is 
.
