HMMT 二月 2004 · 冲刺赛 · 第 20 题

HMMT February 2004 — Guts Round — Problem 20

[8] 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

解析

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: 73 / 9 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 ≥ 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. x y x + y 2( x + y ) − 1