[ Sum of angles in a triangle. Theorem about the exterior angle. ] [ Properties and characteristics of an isosceles triangle. ]

Given an isosceles triangle $A B C$ with vertex $A$. The length of the grasshopper's jump is equal to the base $B C$. It is known that starting from point $C$, the grasshopper reached point $A$ in 22 jumps, landing on the lateral sides of the triangle $A B C$ after each jump and alternating sides with each jump, except for the last one. Find the angles of triangle $A B C$, given that with each jump the grasshopper got closer to point $A$.
Let $A_{0}, A_{1}, A_{2}, \ldots, A_{22}$ be the points visited by the grasshopper ( $A_{0}$ is point $C$, $A_{22}$ is point $A$ ). Before the last jump, the grasshopper was at point $A_{21}$ on side $A B$. If $\angle B A C=\alpha$, then $\angle A_{20} A_{21} B=2 \alpha$ as the external angle of the isosceles triangle $A A_{21} A_{20}$, and since $\angle A_{20} A_{19} A_{21}=\angle A_{20} A_{21} B=2 \alpha$, then $\angle A_{19} A_{20} C=3 \alpha$. Reasoning similarly, we find that $\angle A C B=\angle A B C=\angle B A_{1} C=22 \alpha$. Therefore, $22 \alpha+22 \alpha+\alpha=$ $180^{\circ}$, from which $\alpha=180^{\circ}: 45=4^{\circ}$.

## Answer

$4^{\circ}, 88^{\circ}, 88^{\circ}$.