How many obtuse triangles are there in which the measures of the angles are three different integers?
When arranging the angles in order of magnitude, it is easy to count the triangles, no matter which way they are grouped. Since equal angles cannot appear, $\alpha<\beta<\gamma$ must hold. Since $\gamma \geq 91^{\circ}$, then $\alpha+\beta \leq 89^{\circ}$, so $1^{\circ} \leq \alpha \leq 44^{\circ}$. For every $\alpha$, $\alpha+1 \leq \beta \leq 89-\alpha$, thus $\beta$ can take on a total of $89-2 \alpha$ values, so the number of the sought triangles is:

$$
N=\sum_{\alpha=1}^{44}(89-2 \alpha)=87+85+83+\ldots+5+3+1=\frac{44 \cdot 88}{2}=1936
$$

Gyula Tarnay (St. Imre Grammar School, Budapest, Hungary)