AMC8 2000

AMC8 2000 · Q17

AMC8 2000 · Q17. It mainly tests Manipulating equations, Fractions.

The operation $\otimes$ is defined for all nonzero numbers by $a \otimes b = \frac{a^2}{b}$. Determine $[[(1 \otimes 2) \otimes 3] - [1 \otimes (2 \otimes 3)]]$.

操作$\otimes$对于所有非零数定义为$a\otimes b=\frac{a^2}{b}$。求$[[(1\otimes 2)\otimes 3]-[1\otimes(2\otimes 3)]]$的值。

(A) $-\frac{2}{3}$ $-\frac{2}{3}$

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

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

(E) $\frac{2}{3}$ $\frac{2}{3}$

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): We have $$(1\otimes 2)\otimes 3=\frac{1^2}{2}\otimes 3=\frac{1}{2}\otimes 3=\frac{\left(\frac{1}{2}\right)^2}{3}=\frac{\frac{1}{4}}{3}=\frac{1}{12},$$ and $$1\otimes(2\otimes 3)=1\otimes\left(\frac{2^2}{3}\right)=1\otimes\frac{4}{3}=\frac{1^2}{\frac{4}{3}}=\frac{3}{4}.$$ Therefore, $$(1\otimes 2)\otimes 3-1\otimes(2\otimes 3)=\frac{1}{12}-\frac{3}{4}=\frac{1}{12}-\frac{9}{12}=-\frac{8}{12}=-\frac{2}{3}.$$

答案（A）：我们有 $$(1\otimes 2)\otimes 3=\frac{1^2}{2}\otimes 3=\frac{1}{2}\otimes 3=\frac{\left(\frac{1}{2}\right)^2}{3}=\frac{\frac{1}{4}}{3}=\frac{1}{12},$$ 以及 $$1\otimes(2\otimes 3)=1\otimes\left(\frac{2^2}{3}\right)=1\otimes\frac{4}{3}=\frac{1^2}{\frac{4}{3}}=\frac{3}{4}.$$ 因此， $$(1\otimes 2)\otimes 3-1\otimes(2\otimes 3)=\frac{1}{12}-\frac{3}{4}=\frac{1}{12}-\frac{9}{12}=-\frac{8}{12}=-\frac{2}{3}.$$

Topics