AMC10 2004 A

AMC10 2004 A · Q2

AMC10 2004 A · Q2. It mainly tests Linear equations.

For any three real numbers $a$, $b$, and $c$, with $b \neq c$, the operation $\circledcirc$ is defined by $$\circledcirc(a, b, c) = \frac{a}{b - c}.$$ What is $\circledcirc (\circledcirc(1, 2, 3), \circledcirc(2, 3, 1), \circledcirc(3, 1, 2))$?

对于任意三个实数$a$、$b$和$c$，其中$b \neq c$，操作$\circledcirc$定义为$$\circledcirc(a, b, c) = \frac{a}{b - c}。$$什么是$\circledcirc (\circledcirc(1, 2, 3), \circledcirc(2, 3, 1), \circledcirc(3, 1, 2))$？

(A) -1/2 -$\frac{1}{2}$

(B) -1/4 -$\frac{1}{4}$

(D) 1/4 $\frac{1}{4}$

(E) 1/2 $\frac{1}{2}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

(B) Because $\mathbb{T}(1,2,3)=\dfrac{1}{2-3}=-1,\quad \mathbb{T}(2,3,1)=\dfrac{2}{3-1}=1,\quad \text{and}$ $\mathbb{T}(3,1,2)=\dfrac{3}{1-2}=-3,$ we have $\mathbb{T}\big(\mathbb{T}(1,2,3),\mathbb{T}(2,3,1),\mathbb{T}(3,1,2)\big)=\mathbb{T}(-1,1,-3)$ $=\dfrac{-1}{1-(-3)}=-\dfrac{1}{4}.$

（B）因为 $\mathbb{T}(1,2,3)=\dfrac{1}{2-3}=-1,\quad \mathbb{T}(2,3,1)=\dfrac{2}{3-1}=1,\quad \text{并且}$ $\mathbb{T}(3,1,2)=\dfrac{3}{1-2}=-3,$ 我们有 $\mathbb{T}\big(\mathbb{T}(1,2,3),\mathbb{T}(2,3,1),\mathbb{T}(3,1,2)\big)=\mathbb{T}(-1,1,-3)$ $=\dfrac{-1}{1-(-3)}=-\dfrac{1}{4}.$

Topics

AL.001 Linear equations