2021 AMC 10B Problems/Problem 11
Problem
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?
Solution 1:
Let the side lengths of this rectangular pan be and
; it follows that
. This gives
after some manipulation, so
. By inspection,
maximizes the number of brownies ~ ike.chen
Solution 2:
Let the dimensions of the rectangular pan be and
. The number of interior pieces is
because you cannot include the border, and the number of pieces along the perimeter is
(THIS PART IS FLAWED ~ anonymous user).
Setting these two expressions equal, we have
Applying SFFT (Simon's Favorite Factoring Trick), we get . Doing a bit of trial-and-error, we see that
is maximum when
and
, which gives us a maximum of
brownies.
.
~Bryguy