HMMT 二月 2005 · 代数 · 第 2 题
HMMT February 2005 — Algebra — Problem 2
题目详情
- How many real numbers x are solutions to the following equation? x x x 2003 + 2004 = 2005
解析
- 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 strictly decreasing in x , so there cannot be more than one solution. On the other hand, the left 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.