AMC10 2003 B

AMC10 2003 B · Q16

AMC10 2003 B · Q16. It mainly tests Rounding & estimation, Basic counting (rules of product/sum).

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?

一家餐厅提供三种甜点，小菜的数量恰好是主菜数量的两倍。一顿晚餐包括一道小菜、一道主菜和一道甜点。餐厅应该提供多少最少的主菜数量，使得顾客在2003年每天都能吃到不同的晚餐？

(A) 4 4

(B) 5 5

(D) 7 7

(E) 8 8

Answer

Correct choice: (E)

正确答案：（E）

Solution

(E) Let \(m\) denote the number of main courses needed to meet the requirement. Then the number of dinners available is \(3 \cdot m \cdot 2m = 6m^{2}\). Thus \(m^{2}\) must be at least \(365/6 \approx 61\). Since \(7^{2} = 49 < 61 < 64 = 8^{2}\), 8 main courses is enough, but 7 is not.

（E）令 \(m\) 表示满足要求所需的主菜数量。则可供选择的晚餐数量为 \(3 \cdot m \cdot 2m = 6m^{2}\)。因此 \(m^{2}\) 至少应为 \(365/6 \approx 61\)。由于 \(7^{2}=49<61<64=8^{2}\)，所以需要 8 道主菜才足够，而 7 道不够。

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