1986 AIME 第 2 题

Question

计算乘积

\begin{gathered} (\sqrt5+\sqrt6+\sqrt7)\\ {}\cdot(-\sqrt5+\sqrt6+\sqrt7)\\ {}\cdot(\sqrt5-\sqrt6+\sqrt7)\\ {}\cdot(\sqrt5+\sqrt6-\sqrt7) \end{gathered}\text{。}

Evaluate the product

\begin{gathered} (\sqrt5+\sqrt6+\sqrt7)\\ {}\cdot(-\sqrt5+\sqrt6+\sqrt7)\\ {}\cdot(\sqrt5-\sqrt6+\sqrt7)\\ {}\cdot(\sqrt5+\sqrt6-\sqrt7). \end{gathered}

答案

将因式配对，使每一对都可用平方差化简

Pair factors so that each pair is a difference of squares

第一次配对后，剩余的根式只会以

\sqrt{42}

的形式出现

After the first pairing, the remaining radicals occur only through

\sqrt{42}

先将前两个因式配对，再将后两个配对：

\begin{gathered} (\sqrt6+\sqrt7)^2-(\sqrt5)^2\\ {}=8+2\sqrt{42},\\ (\sqrt5)^2-(\sqrt7-\sqrt6)^2\\ {}=-8+2\sqrt{42} \end{gathered}\text{。}

因此所求乘积为

\begin{aligned} &(8+2\sqrt{42})(-8+2\sqrt{42})\\ &\qquad=168-64=104 \end{aligned}\text{。}

Pair the first two factors, then the last two:

\begin{gathered} (\sqrt6+\sqrt7)^2-(\sqrt5)^2\\ {}=8+2\sqrt{42},\\ (\sqrt5)^2-(\sqrt7-\sqrt6)^2\\ {}=-8+2\sqrt{42}. \end{gathered}

Therefore the requested product is

\begin{aligned} &(8+2\sqrt{42})(-8+2\sqrt{42})\\ &\qquad=168-64=104. \end{aligned}

其他年份的第 2 题