Problem 7-4. In triangle $A B C$, where all three sides are pairwise distinct, the angle bisectors of angles $A$ and $B$ are drawn, dividing it into a quadrilateral and three triangles, two of which are isosceles. Find the angles of the original triangle.
Answer: $4 \pi / 7, 2 \pi / 7, \pi / 7$.

Solution. Let in triangle $ABC$ the bisectors $AK$ and $BL$ intersect at point $O, \angle CAB = \alpha, \angle CBA = \beta$. Triangle $ABO$ cannot be isosceles, as none of the following conditions can be satisfied: $\alpha = \beta, \alpha + \beta / 2 = 180^{\circ}, \beta + \alpha / 2 = 180^{\circ}$. We will find under what condition triangle $ALO$ will be isosceles. The equality $\angle LAO = \angle LOA$ is impossible, since $\angle LAO = \angle BAO$, and lines $BL$ and $AB$ are not parallel. The equality $\angle LAO = \angle LOA$ is equivalent to $\beta + 3\alpha / 2 = 180^{\circ}$. Similarly, the equality $\angle LAO = \angle ALO$ is equivalent to $\beta + 3\alpha = 360^{\circ}$. Then, according to the problem's condition, one of the following pairs of equalities is satisfied:

$$
\begin{gathered}
\beta + 3\alpha / 2 = 180^{\circ}, \quad \alpha + 3\beta / 2 = 180^{\circ} \\
\beta + 3\alpha = 360^{\circ}, \quad \alpha + 3\beta = 360^{\circ} \\
\beta + 3\alpha / 2 = 180^{\circ}, \quad \alpha + 3\beta = 360^{\circ} \\
\beta + 3\alpha = 360^{\circ}, \quad \alpha + 3\beta / 2 = 180^{\circ}
\end{gathered}
$$

But the first pair of equalities cannot be satisfied simultaneously, as $\alpha \neq \beta$. And the second pair of equalities cannot be satisfied, as $\alpha + \beta \neq 180^{\circ}$. The third and fourth pairs of equalities lead to the specified answer.

| Rating | Score | Content criterion |
| :---: | :---: | :--- |
| + | 20 | Complete solution |
| .+ | 18 | Fully correct solution, but an arithmetic error or minor flaw is made |
| $+/-$ | 15 | Isosceles property is proven, but not all equations are correctly formulated, leading to incorrect angles |
| $+/ 2$ | 10 | Isosceles property is not proven, but equations and answer are correct |
| $-/+$ | 6 | Isosceles property is not proven, but correct equations are obtained |
| .- | 4 | Proven that the central triangle is not isosceles |
| - | 0 | Solution is completely incorrect/only answer |