1994 AIME Problem 12

Question

A fenced, rectangular field measures

24

meters by

52

meters. An agricultural researcher has

1994

meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partitioned using all or some of the

1994

meters of fence?

Answer

Answer: 702

Concepts:tiling divisibility optimization

Difficulty rating: 1900

Small hint:

If there are

m

rows and

n

columns, equality of square side lengths forces

m:n=6:13

Big hint:

Write

m=6k

and

n=13k

, then calculate only the internal fence length

Solution:

Let the grid have

6k

rows and

13k

columns, so each square has side

\frac{4}{k}.

The internal vertical and horizontal fences have total length

\begin{aligned}&52(6k-1)+24(13k-1)\\&\quad=624k-76.\end{aligned}

Requiring this to be at most

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gives

k\leq3.

With

k=3,

the number of plots is

(6k)(13k)=78k^2=702.