1. A Pythagorean triangle is a right-angled triangle where all three sides are integers. The most famous example is the triangle with legs 3 and 4 and hypotenuse 5.

Determine all Pythagorean triangles for which the area is equal to twice the perimeter.
1. For a Pythagorean triangle with legs $a$ and $b$ and hypotenuse $c$, the following holds: $a^{2}+b^{2}=c^{2}$. The area is equal to $\frac{1}{2} a b$ and the perimeter is equal to $a+b+c$.

Thus $\frac{1}{2} a b=2(a+b+c)$ or $\frac{a b}{4}=a+b+c$ so $\frac{a b}{4}-a-b=c$

$a^{2}+b^{2}=\left(\frac{a b}{4}-a-b\right)^{2}, \quad 0=\left(\frac{a b}{4}\right)^{2}-\frac{a^{2} b}{2}-\frac{a b^{2}}{2}+2 a b, \quad a b-8 a-8 b+32=0$.

The last expression can be rewritten as $a b-8 a-8 b+64=32$ so $(a-8)(b-8)=32$. Since 32 can be factored in three ways $1 \times 32, 2 \times 16, 4 \times 8$, we find three solutions for $a$ and $b$ where the corresponding $c$ can be easily determined: $(9,40,41),(10,24,26)$ and $(12,16,20)$.