AMC10 2021 B

AMC10 2021 B · Q11

AMC10 2021 B · Q11. It mainly tests Linear equations, Area & perimeter.

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?

奶奶刚刚烤好了一大块长方形布朗尼。她计划切成相同大小和形状的长方形小块，使用平行于锅边的直切。每刀必须完全横切锅。奶奶希望内部小块的数量与锅边缘小块的数量相同。她能制作的最大布朗尼数量是多少？

(A) 24 24

(B) 30 30

(D) 60 60

(E) 64 64

Answer

Correct choice: (D)

正确答案：（D）

Solution

Let the side lengths of the rectangular pan be $m$ and $n$. It follows that $(m-2)(n-2) = \frac{mn}{2}$, since half of the brownie pieces are in the interior. This gives $2(m-2)(n-2) = mn \iff mn - 4m - 4n + 8 = 0$. Adding $8$ to both sides and applying Simon's Favorite Factoring Trick, we obtain $(m-4)(n-4) = 8$. Since $m$ and $n$ are both positive, we obtain $(m, n) = (5, 12), (6, 8)$ (up to ordering). By inspection, $5 \cdot 12 = \boxed{\textbf{(D) }60}$ maximizes the number of brownies.

设长方形锅的边长为$m$和$n$。内部小块数为$(m-2)(n-2)$，总小块数为$mn$，边缘小块数为$mn - (m-2)(n-2)$。题目要求$(m-2)(n-2) = mn/2$，即$2(m-2)(n-2) = mn \iff mn - 4m - 4n + 8 = 0$。两边加8并应用Simon的最爱因式分解技巧，得$(m-4)(n-4) = 8$。由于$m,n$为正整数，得到$(m,n)=(5,12),(6,8)$（顺序交换）。检验，$5 \cdot 12 = \boxed{\textbf{(D) }60}$最大化布朗尼数量。

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