PUMaC 2019 · 代数（A 组） · 第 1 题

PUMaC 2019 — Algebra (Division A) — Problem 1

Let x and y be positive real numbers that satisfy (log x ) + (log y ) = log( x ) + log( y ). 2 Compute the maximum possible value of (log xy ) . 2

解析

Let x and y be positive real numbers that satisfy (log x ) + (log y ) = log( x ) + log( y ). 2 Compute the maximum possible value of (log xy ) . Proposed by: Matthew Kendall Answer: 16 2 2 2 Let u = log x and v = log y . Then u + v = 2 u + 2 v . Completing the square gives ( u − 1) + √ 2 ( v − 1) = 2, so the equation given is a circle of radius 2 centered at (1 , 1) on the uv plane. 2 Let log xy = u + v = k , so we wish to maximize k . Note that the line u + v is tangent to the √ circle when the origin is a distance of 0 or 2 2 from the line. The latter gives u = v = 2, so 2 k = 4, making the maximum k = 16 . 2