Question

奶奶刚烤好一大盘长方形布朗尼。她打算沿着盘子的边平行地直线切开，做成大小和形状相同的长方形小块。每一刀都必须从盘子的一边完全切到另一边。奶奶希望内部小块的数量与位于周边的小块数量相同。她最多可以做出多少块布朗尼？

Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce?

$24$

$30$

$48$

$60$

$64$

答案：D

知识点：西蒙最爱的因式分解技巧最优化

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文字解答：

假设切成 $l\times w$ 的网格。内部小块数为 $(l-2)(w-2)$ ，总块数为 $lw$ 。

$(l-2)(w-2)=\frac{lw}{2}.$

展开得到 $lw-4l-4w+8=0$ ，整理为：

$(l-4)(w-4)=8.$

$8$ 的正因数对给出 $(l,w)=(5,12)$ 或 $(6,8)$ ，顺序可交换。它们分别产生 $60$ 或 $48$ 块，所以最大可能数量是 $60$ 。

所以答案是 D。

Suppose the cuts make an $l\times w$ grid of pieces. The number of interior pieces is $(l-2)(w-2),$ and the total number of pieces is $lw.$ Since the number of interior pieces equals the number of perimeter pieces, the interior pieces make up half the total:

$(l-2)(w-2)=\frac{lw}{2}.$

Multiplying out gives $lw-4l-4w+8=0,$ or

$(l-4)(w-4)=8.$

The positive factor pairs of $8$ give $(l,w)=(5,12)$ or $(6,8),$ up to order. These produce $60$ or $48$ pieces, respectively, so the greatest possible number is $60.$

Thus, the answer is D .

2021 AMC 10B Spring 第 11 题

答案：D

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其他年份的第 11 题