Question

A rectangle is called cool if the number of square units in its area is equal to twice the number of units in its perimeter. A cool rectangle also must have integer side lengths. What is the sum of all the different possible areas of cool rectangles?

Answer 1

Rectangles that are 5×20, 6×12, or 8×8 will satisfy your criteria. The sum of the areas of those is 236 units².

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20×5 and 12×6 rectangles are also "cool", but their areas are not different from those already counted.