Consider those triangles in which the measure of each side is a natural number not greater than 10. What is the largest possible angle and the smallest possible angle that can occur in these triangles?
Let's choose the labeling of the sides such that $a \geq b \geq c$. Then for the angles, $\alpha \geq \beta \geq \gamma$.
I. $\gamma$.

For fixed $a$ and $u$, the expression is the largest when $v$ is the smallest possible, i.e., $v=0$ if $u$ is even, and $v=1$ if $u$ is odd. In the first case,

$$
\frac{u^{2}-v^{2}}{u^{2}-a^{2}}=\frac{1}{1-\left(\frac{a}{u}\right)^{2}}
$$

and this is the largest when $a / u$ is the largest. For this, allowing $u$ to vary, $u$ must be the smallest. Due to the triangle inequality, $a81 / 19$, so the value of $\operatorname{tg}^{2} \alpha / 2$ and thus $\alpha$ is the largest when $a=10, u=11, v=1, b=6, c=5$. In this case, $\alpha_{\max } \approx 130.5^{\circ}$.

Tibor Faragó (Budapest, Bláthy O. Erősár. Ip. Tech., IV. o. t.)