In triangle $ABC$, the measure of each angle in degrees is an integer. At vertex $A$, there is an obtuse angle. Point $D$ is the midpoint of side $BC$, such that triangle $ABD$ is a right triangle, and triangle $ADC$ is isosceles. - How many such triangular shapes are there? (It is not necessary to list the angles of all the suitable shapes.)





Translating the text into English while preserving the original line breaks and formatting, the result is as follows.
Given that $\angle ABC = \beta$ and $\angle ACB = \gamma$ are acute angles, the right angle in triangle $ABD$ is either at $A$ or at $D$.

If $\angle BAD = 90^\circ$, then by the exterior angle theorem, $\angle ADC = 90^\circ + \beta > 90^\circ$. Therefore, the equal angles of triangle $ADC$ are at $A$ and $C$, $\gamma = \angle CAD = 45^\circ - \beta / 2 = 45^\circ$, and thus $\beta < 45^\circ$. In this case, $\beta$ can take the values $1^\circ, 2^\circ, \ldots, 44^\circ$, giving us 44 different triangles.

There is no triangle that we have counted twice, so the number of triangles with the required properties is 88.

Sain Ildikó (Budapest, Apáczai Csere J. Gyak. Gimn. I. o. t.)