HMMT 二月 2000 · ORAL 赛 · 第 2 题
HMMT February 2000 — ORAL Round — Problem 2
题目详情
英文原题
- [30] How many positive solutions are there to x + 7 x + 14 x + 1729 x − 1379 x = 0 ?
How many positive integer solutions ?
n n n
解析
英文解析
- Dividing by x , we get the equation x + 7 x + 14 x + 1729 x − 1379 = 0. This is amonotonically increasing function of x on [0 , ∞ ). From f (0) < 0 and f ( x ) → ∞ asx → ∞ we see that there’s only one positive solution. In fact, f (1) > 0, so there’s nopositive integer solution.
n n n − 1 n − 1 n − 1