|
|
| (11 intermediate revisions by 4 users not shown) |
| Line 1: |
Line 1: |
| − | ==Problem==
| + | #REDIRECT [[2021_Fall_AMC_12B_Problems/Problem_5]] |
| − | | |
| − | Call a fraction <math>\frac{a}{b}</math>, not necessarily in the simplest form special if <math>a</math> and <math>b</math> are positive integers whose sum is <math>15</math>. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
| |
| − | | |
| − | <math>\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\
| |
| − | 12 \qquad\textbf{(E)}\ 13</math>
| |
| − | | |
| − | ==Solution==
| |
| − | | |
| − | Listing out all special fractions, we get:
| |
| − | <math>\{\frac{1}{14}, \frac{2}{13}, \frac{3}{12}, \frac{4}{11}, \frac{5}{10}, \frac{6}{9}, \frac{7}{8}, \frac{8}{7}, \frac{9}{6}, \frac{10}{5}, \frac{11}{4}, \frac{12}{3}, \frac{13}{2}, \frac{14}{1}</math>}
| |
| − | | |
| − | Simplifying and grouping based on their denominators gives
| |
| − | | |
| − | <math>\{14, 4, 2</math>}<math>, \{\frac{1}{2}, \frac{3}{2}, \frac{13}{2}</math>}<math>, \{\frac{1}{4}, \frac{11}{4}</math>}
| |
| − | | |
| − | ==See Also==
| |
| − | {{AMC10 box|year=2021 Fall|ab=B|num-a=8|num-b=6}}
| |
| − | {{MAA Notice}}
| |
