3. Denote, that there exist such points $D$ on side $A B$ and $E$ on side $A C$ of triangle $A B C$, such that $|A E|=|E D|=|D B|$ and $|A D|=|D C|=|C B|$. Determine the sizes of the angles of triangle $A B C$.
$\mathrm{I} / 3$.

![](https://cdn.mathpix.com/cropped/2024_06_07_8c492ac180897ee339acg-19.jpg?height=534&width=743&top_left_y=470&top_left_x=271)

Let $\angle E B D=\alpha$. Then $\angle D E B=\alpha$, so $\angle E D A=2 \alpha$ and $\angle D A E=2 \alpha$. It follows that $\angle D E C=4 \alpha$ or $\angle B E C=3 \alpha$. At the same time, $\angle A C D=\angle D A C=2 \alpha$, so $\angle B D C=4 \alpha$ and $\angle C B D=4 \alpha$ or $\angle C B E=3 \alpha$. It follows that triangle $E B C$ is isosceles with the vertex at $C$, so $|C E|=|C D|$ and therefore $\angle C D E=\angle D E C=4 \alpha$. Thus,

$$
180^{\circ}=\angle B D C+\angle C D E+\angle E D A=4 \alpha+4 \alpha+2 \alpha=10 \alpha
$$

or $\alpha=18^{\circ}$. From this, we calculate $\angle B A C=2 \alpha=36^{\circ}, \angle C B A=4 \alpha=72^{\circ}$ and $\angle A C B=$ $180^{\circ}-\angle B A C-\angle C B A=72^{\circ}$.

## Let $\angle E B D=\alpha$.

![](https://cdn.mathpix.com/cropped/2024_06_07_8c492ac180897ee339acg-19.jpg?height=60&width=1633&top_left_y=1506&top_left_x=220)

Observation $\angle E D A=\angle D A E=\angle A C D=2 \angle E B D=2 \alpha \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_8c492ac180897ee339acg-19.jpg?height=57&width=1633&top_left_y=1605&top_left_x=220)

![](https://cdn.mathpix.com/cropped/2024_06_07_8c492ac180897ee339acg-19.jpg?height=51&width=1631&top_left_y=1659&top_left_x=218)

![](https://cdn.mathpix.com/cropped/2024_06_07_8c492ac180897ee339acg-19.jpg?height=52&width=1633&top_left_y=1710&top_left_x=220)

![](https://cdn.mathpix.com/cropped/2024_06_07_8c492ac180897ee339acg-19.jpg?height=54&width=1631&top_left_y=1755&top_left_x=218)

Calculation $\angle B A C=36^{\circ}, \angle C B A=72^{\circ}$ and $\angle A C B=72^{\circ} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 1$ point

The problem can be solved in many ways; another version of the scoring guide is provided:

Let $\angle B A C=\alpha$ and $\angle C B A=\beta$. Observation $2 \alpha=\beta \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$.

Observation $\triangle E B C$ is isosceles or quadrilateral $C E D B$ is a kite . . . . . . . 3 points

Calculation of angles $\angle B A C=36^{\circ}, \angle C B A=72^{\circ}$ and $\angle A C B=72^{\circ} \ldots \ldots \ldots \ldots \ldots \ldots$.

If a participant reaches a conclusion without any reasoning, proof, or calculation, 2 points are deducted. For each further unjustified conclusion, 1 point is deducted.