11.5. The lateral faces of the pentagonal pyramid $S A B C D E$ are acute-angled triangles. We will call a lateral edge of the pyramid good if it is equal to the height of the opposite lateral face, drawn from the vertex of the pyramid (for example, the edge $S A$ is good if it is equal to the height of the triangle $S C D$, drawn from the vertex $S$). What is the maximum number of good edges that the pyramid can have?
# Answer. 2.

Solution. We will show that two non-adjacent lateral edges of the pyramid cannot both be good. Suppose this is not the case, and, for example, edges $S A$ and $S C$ are good, meaning $S A = S P$ is the height of face $S C D$, and $S C = S Q$ is the height of face $S A E$. However, by the property of the height in an acute triangle, $S P < S C$, and $S Q < S A$. Therefore, $S A = S P < S C = S Q < S A$ - a contradiction.

Now, note that among the three lateral edges of a pentagonal pyramid, there will always be two non-adjacent edges. Thus, the pyramid can have no more than two good edges. We will show that there exists a pentagonal pyramid with two good lateral edges. Consider a square $A M N E$ with center $O$. Let the line passing through vertex $M$ and perpendicular to $E M$, and the line passing through vertex $N$ and perpendicular to $A N$, intersect at point $C$. Choose points $B$ and $D$ such that points $M$ and $N$ are the midpoints of segments $B C$ and $D C$, respectively. Choose an arbitrary point $S$ on the perpendicular to the plane $A B C$ passing through point $O$. Then, by the theorem of three perpendiculars, $S M$ and $S N$ are the heights of triangles $S B C$ and $S D C$. And, as follows from the isosceles nature of triangles $E S M$ and $A S N$, edges $S E$ and $S A$ are good.

Comment. Correct answer without justification - 0 points.

Proved that the pyramid cannot have more than 2 good edges - 4 points.

Provided an example of a pyramid with 2 good edges - 3 points.