## Task A-3.6.

Let $\overline{BD}$ and $\overline{CE}$ be the altitudes of an acute-angled triangle $ABC$. Determine the smallest measure of angle $\varangle BAC$ for which it is possible that $|AE| \cdot |AD| = |BE| \cdot |CD|$.
Scoring: To fully solve the problem, it is necessary to show that equality can be achieved for the value $60^{\circ}$. This part of the solution is worth 2 points.

The claim that $\cos \varangle B A C \leqslant \frac{1}{2}$, or that $\alpha \geqslant 60^{\circ}$, is worth 2 points. The proof of this claim is worth 6 points, of which 4 points are for expressing $\cos \varangle B A C$ in terms of $|A B|$ and $|A C|$. Students can prove that $\frac{b c}{b^{2}+c^{2}} \leqslant \frac{1}{2}$ in multiple ways, for example, by using the A-G inequality, and each correct proof is worth 2 points.

Applying trigonometry to different right triangles, without further conclusions, is worth a maximum of 1 point.