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2021 AMC 10B Problem 11

Problem 11 of 25IntermediateAlgebraProblem-Solving Techniques

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?

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Solution

Suppose the cuts make an l×wl\times w grid of pieces. The number of interior pieces is (l−2)(w−2),(l-2)(w-2), and the total number of pieces is lw.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)=lw2.(l-2)(w-2)=\frac{lw}{2}. Multiplying out gives lw−4l−4w+8=0,lw-4l-4w+8=0, or (l−4)(w−4)=8.(l-4)(w-4)=8. The positive factor pairs of 88 give (l,w)=(5,12)(l,w)=(5,12) or (6,8),(6,8), up to order. These produce 6060 or 4848 pieces, respectively, so the greatest possible number is 60.60. Thus, the answer is D .
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