Question

一家餐厅提供三种甜点，前菜数量正好是主菜数量的两倍。一顿晚餐由一道前菜、一道主菜和一道甜点组成。餐厅至少应提供多少种主菜，才能让顾客在 $2003$ 年每天晚上都吃不同的晚餐？

A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that the restaurant should offer so that a customer could have a different dinner each night in the year $2003?$

$4$

$5$

$6$

$7$

$8$

答案：E

知识点：乘法原理不等式

难度评级：1370

解答：

若有 $m$ 种主菜，则前菜种数是主菜的两倍，晚餐总数为 $3 \cdot m \cdot 2m = 6m^2$ 。这个数至少需要达到 $365$ 。

$m^2 \ge \dfrac{365}{6}\approx 60.8$ 。因为 $7^2=49$ 太小，而 $8^2=64$ 可行，所以 $m=8$ 。

所以正确答案是 E。

With $m$ main courses, the number of dinners is $3 \cdot m \cdot 2m = 6m^2.$ This must be at least $365.$

So $m^2 \ge \dfrac{365}{6}\approx 60.8.$ Since $7^2=49$ is too small but $8^2=64$ works, $m=8.$

Thus, the correct answer is E.

2003 AMC 10B 第 16 题

答案：E

难度评级：1370

解答：

其他年份的第 16 题