PRMO 2015 question 9

9. A \(2 \times 3\) rectangle and a \(3 \times 4\) rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?

Solution:

There are 2 optimal ways to arrange the rectangles. 
First do it vertically like this so that '2' and '4' are aligned.
2x3
4x3
So it becomes 6x3
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The smallest square containing this has to be 6x6 with area 36.

Second, '2' and '3' are aligned:
2x3
3x4

Now it becomes roughly 5x4 and the surrounding square has to be 5x5 = 25.
Answer: 25

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