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Harvard-MIT Mathematics Tournament

Algebra Difficulty 4.9 AIME Prove it United States

Problem:

Two positive rational numbers $x$ and $y$ , when written in lowest terms, have the property that the sum of their numerators is $9$ and the sum of their denominators is $10$ . What is the largest possible value of $x + y$ ?

Solution

Solution:

For fixed denominators $a < b$ (with sum $10$ ), we maximize the sum of the fractions by giving the smaller denominator as large a numerator as possible: $8/a + 1/b$ . Then, if $a \geq 2$ , this quantity is at most $8/2 + 1/1 = 5$ , which is clearly smaller than the sum we get by setting $a = 1$ , namely $8/1 + 1/9 = 73/9$ . So this is the answer.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.