+86 A right triangle with integer leg lengths is called cool if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles?
+9665 Suppose the legs are a and b. Then by the definition of cool trianlges,
\(\dfrac{ab}2 = 2(a + b)\\ ab = 4a + 4b\\ ab - 4a - 4b + 16 = 16\\ (a - 4)(b - 4) = 16\)
Assume without loss of generality that \(a \leq b\). Then
\(\begin{cases} a-4=1\\ b-4=16 \end{cases}\text{ or }\begin{cases} a-4=2\\ b-4=8 \end{cases}\text{ or }\begin{cases} a-4=4\\ b-4=4 \end{cases} \)
This gives all three cool triangles, the one with legs 5 and 20, the one with legs 6 and 12, and the one with both legs equal to 8.
The sum of all possible areas is \(\dfrac{5 \times 20 + 6 \times 12 + 8 \times 8}2 = 118\).
