HMMT 二月 2019 · ALGNT 赛 · 第 3 题

HMMT February 2019 — ALGNT Round — Problem 3

Let x and y be positive real numbers. Define a = 1 + and b = 1 + . If a + b = 15, compute a + b . y x

解析

Let x and y be positive real numbers. Define a = 1 + and b = 1 + . If a + b = 15, compute a + b . y x Proposed by: Michael Tang Answer: 50 y x Note that a − 1 = and b − 1 = are reciprocals. That is, y x ( a − 1)( b − 1) = 1 = ⇒ ab − a − b + 1 = 1 = ⇒ ab = a + b. Let t = ab = a + b . Then we can write 2 2 2 2 a + b = ( a + b ) − 2 ab = t − 2 t, 2 so t − 2 t = 15, which factors as ( t − 5)( t + 3) = 0. Since a, b > 0, we must have t = 5. Then, we compute 3 3 3 3 2 a + b = ( a + b ) − 3 ab ( a + b ) = 5 − 3 · 5 = 50 .