AIME 2000 I · 第 3 题

AIME 2000 I — Problem 3

专题: Contest Math
难度: L4
来源: AIME

题目详情

Problem

In the expansion of $(ax + b)^{2000},$ where $a$ and $b$ are relatively prime positive integers, the coefficients of $x^{2}$ and $x^{3}$ are equal. Find $a + b$ .

解析

Solution

Using the binomial theorem, $\binom{2000}{1998} b^{1998}a^2 = \binom{2000}{1997}b^{1997}a^3 \Longrightarrow b=666a$ .

Since $a$ and $b$ are positive relatively prime integers, $a=1$ and $b=666$ , and $a+b=\boxed{667}$ .