2021 AMC 12B Fall Problem 5

Question

Call a fraction

\dfrac{a}{b},

not necessarily in the simplest form, special if

a

and

b

are positive integers whose sum is

15.

How many distinct integers can be written as the sum of two, not necessarily different, special fractions?

9

10

11

12

13

List the special fractions and note which are integers, half-integers, or quarter-integers

Two specials sum to an integer only when their fractional parts cancel

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.

Problem 5 in other years