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
.