HMMT 二月 2005 · 代数 · 第 2 题

英文解析

2. How many real numbers x are solutions to the following equation?
   x x x
   2003 + 2004 = 2005
   Solution: 1
   x x
   Rewrite the equation as (2003 / 2005) + (2004 / 2005) = 1. The left side is strictlydecreasing in x , so there cannot be more than one solution. On the other hand, theleft side equals 2 > 1 when x = 0 and goes to 0 when x is very large, so it must equal
   1 somewhere in between. Therefore there is one solution.