3. Let $A B C$ be a right triangle with a right angle at $B$. On the side $B C$ lie points $D$ and $E$ such that $\angle B A D = \angle D A E = \angle E A C$ and $|E C| = 2|B D|$. Determine the angles of triangle $A B C$.
IV/3. Let $\alpha=\angle B A C$. Complete the triangle $A B C$ to a rectangle $A B C F$, and then reflect the segment $B C$ over the segment $A F$ to the segment $B^{\prime} C^{\prime}$. Since $|E C|=2|B D|$, we have $\angle E C^{\prime} C = \angle B A D = \alpha / 3 = \angle C A E$. Therefore, the quadrilateral $A E C C^{\prime}$ is cyclic, and thus $\angle A C E = \angle A C^{\prime} E = \pi / 2 - \alpha$. Hence, $2(\pi / 2 - \alpha) + \alpha / 3 = \angle B^{\prime} C^{\prime} C = \pi / 2$. It follows that $\angle B A C = \alpha = 3 \pi / 10$ and $\angle A C B = \pi / 5$.

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Introduction of rectangles $A B C F$ and $B^{\prime} B C C^{\prime}$: 1 point each. Cyclic nature of quadrilateral $A E C C^{\prime}$: 2 points. Equation $2(\pi / 2 - \alpha) + \alpha / 3 = \pi / 2$: 2 points. Conclusion $\alpha = 3 \pi / 10$ and $\angle A C B = \pi / 5$: 1 point. If the contestant approaches the problem using trigonometry but does not solve it completely: derivation of a correct and solvable equation for one unknown angle: 4 points; if no appropriate equation is derived for the angles: each correct observation: 1 point, but a maximum of 3 points in total.