2021 Fall AMC 12B Problem 5

Listing $\frac{a}{15-a}$ for $1\le a\le14,$ the integers are $2,4,14;$ the fractions with fractional part $\frac{1}{2}$ are $\tfrac12,\tfrac32,\tfrac{13}{2};$ and $\tfrac14,\tfrac{11}{4}$ have complementary fractional parts $\frac{1}{4},\frac{3}{4}.$ The remaining fractional parts are $\frac{1}{14},\frac{2}{13},\frac{4}{11},\frac{2}{3},\frac{7}{8},\frac{1}{7},$ and none of their complements occurs. Thus only the integer, half-integer, and quarter cases can give integer sums. Integer pairs give $4, 6, 8, 16, 18, 28.$ Half-integer pairs $\left(\tfrac12, \tfrac32, \tfrac{13}{2}\right)$ give $1, 2, 3, 7, 8, 13.$ The quarter pair $\tfrac14 + \tfrac{11}{4}$ gives $3.$ The distinct integers are $1, 2, 3, 4, 6, 7, 8, 13, 16, 18, 28,$ a total of $11.$ Thus, the correct answer is C.