A triangle is given. If we shorten its longest side by one third of its length, double the shortest side, and decrease the remaining side by $2 \mathrm{~cm}$, we get a triangle congruent to the original triangle. What are the dimensions of this triangle?

(M. Raabová)
Let's denote the sides of the triangle as $a, b, c$, and we can assume that $a>b>c$. From the problem statement, it follows that I reduce one side by $2 \mathrm{~cm}$, so I must get side $c$, i.e., $b-2=c$. Furthermore, it is given that $\frac{2}{3}=b$ and $2 c=a$. The only solution to this system is $a=12, b=8, c=6$.