2023 AIME II Problems/Problem 5
Problem
Let be the set of all positive rational numbers such that when the two numbers and are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of can be expressed in the form where and are relatively prime positive integers. Find
Solution
Denote , where . We have . Suppose , then the sum of the numerator and the denominator of is . This cannot be equal to the sum of the numerator and the denominator of , . Therefore, .
Case 1: can be written as with .
Thus, .
Because the sum of the numerator and the denominator of and are the same,
Hence, .
Because , . Thus, and . Therefore, .
Case 2: can be written as with .
Thus, .
Because the sum of the numerator and the denominator of and are the same,
Hence, .
Because , . Thus, and . Therefore, .
Case 3: can be written as .
Thus, .
Because the sum of the numerator and the denominator of and are the same,
Hence, . This is infeasible. Thus, there is no solution in this case.
Putting all cases together, . Therefore, the sum of all numbers in is
Therefore, the answer is .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution by The Power of Logic
See also
2023 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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