2021 AMC 10 B Fall
Complete problem set with solutions and individual problem pages
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?(2021 AMC Fall 10B, Question #7)
- A.
- B.
- C.
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- E.
Solution 1:
The special fractions are We rewrite them in the simplest form: Note that two unlike fractions in the simplest form cannot sum to an integer. So, we only consider the fractions whose denominators appear more than once: For the set , two elements (not necessarily different) can sum to . For the set , two elements (not necessarily different) can sum to .
For the set , two elements (not necessarily different) can sum to 3 . Together, there are (C) 11 distinct integers that can be written as the sum of two, not necessarily different, special fractions:
Solution 2:
Let , so the special fraction is We can ignore the part and only focus on . The integers are , which are , respectively. We get from this group of numbers.
The halves are , which are , respectively. We get from this group of numbers. The quarters are , which are , respectively. We get 5 from this group of numbers. Note that 10 and 5 each appear twice. Therefore, the answer is (C) 11 .
