HMMT 二月 2004 · 冲刺赛 · 第 20 题

英文解析

20. Two positive rational numbers x and y , when written in lowest terms, have the propertythat the sum of their numerators is 9 and the sum of their denominators is 10. Whatis 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 bygiving the smaller denominator as large a numerator as possible: 8 /a + 1 /b . Then, ifa ≥ 2, this quantity is at most 8 / 2 + 1 / 1 = 5, which is clearly smaller than the sumwe 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