Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Length and Inequalities.
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, find the largest number of square test plots into which the field can be partitioned using all or some of the 1994 meters of fence.
Integers
Inequalities
Length
Answer: is 702.
AIME I, 1994, Question 12
Inequalities (Little Mathematical Library) by Korovkin
Number of squares in a row=\(\frac{52n}{24}\)=\(\frac{13n}{6}\) squares in every row
each vertical fence lengths 24 for \(\frac{13n}{6}-1\) vertical fences
each horizontal fence lengths 52 for n-1 such fences
total length of internal fencing 24 (\(\frac{13n}{6}-1\))+52(n-1)=104n-76 \( \leq 1994\)
\(\Rightarrow n \leq \frac{1035}{52}\)
\(\Rightarrow n \leq 19\)
the largest multiple of 6 that is \( \leq 19 \)
\(\Rightarrow n=18\)
required number =\(\frac{13n^{2}}{6}\)=702.
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