11.4. Consider all acute-angled triangles with a given side $a$ and angle $\alpha$. What is the maximum value of the sum of the squares of the lengths of sides $b$ and $c$?
11.4. By the Law of Cosines, $b^{2}+c^{2}=a^{2}+2 b c \cos \alpha$. Since $2 b c \leqslant b^{2}+c^{2}$ and $\cos \alpha>0$, then $b^{2}+c^{2} \leqslant a^{2}+\left(b^{2}+c^{2}\right) \cos \alpha$, i.e., $b^{2}+c^{2} \leqslant a^{2} /(1-\cos \alpha)$. Equality is achieved if $b=c$.