\section*{Problem 3A - 141043A}

Let \(A B C D S\) be a straight quadrilateral pyramid with a fixed square base \(A B C D\).

We consider all closed line segments \(P Q R T P\), where \(P\) is a fixed interior point of the edge \(A S\), \(Q\) is an interior point of \(B S\), \(R\) of \(C S\), and \(T\) of \(D S\).

Determine the set of all angles \(\rho\left(0^{\circ}<\rho<90^{\circ}\right)\) for which the following holds:

If the angle \(\angle A S B\) has the size \(\rho\), then among the line segments \(P Q R T P\) considered on the pyramid \(A B C D S\), there exists a shortest one.
We imagine the pyramid's mantle cut along the edges \(S A\) and unfolded into a plane to form a figure as shown in the diagram, a hexagon \(S^{\prime} A^{\prime} B^{\prime} C^{\prime} D^{\prime} A^{\prime \prime}\), where

\[
\triangle S^{\prime} A^{\prime} B^{\prime} \cong \triangle S A B \quad ; \quad \triangle S^{\prime} B^{\prime} C^{\prime} \cong \triangle S B C \quad ; \quad \triangle S^{\prime} C^{\prime} D^{\prime} \cong \triangle S C D \quad ; \quad \triangle S^{\prime} D^{\prime} A^{\prime \prime} \cong \triangle S D A
\]

and for the interior angle \(\angle A^{\prime} S^{\prime} A^{\prime \prime}\) of the hexagon, \(\angle A^{\prime} S^{\prime} A^{\prime \prime}=4 \varphi\) holds. Furthermore:

(1) Each of the closed line segments \(P Q R T P\) is transformed into an equally long non-closed line segment \(P^{\prime} Q^{\prime} R^{\prime} T^{\prime} P^{\prime \prime}\), where \(P^{\prime}, Q^{\prime}, R^{\prime}, T^{\prime}, P^{\prime \prime}\) are internal points of \(S A^{\prime}, S B^{\prime}\), \(S C^{\prime}, S D^{\prime}, S A^{\prime \prime}\) respectively, and \(S^{\prime} P^{\prime}=S^{\prime} P^{\prime \prime}\) holds.

Conversely, each of the line segments \(P^{\prime} Q^{\prime} R^{\prime} T^{\prime} P^{\prime \prime}\) mentioned in (1) corresponds to an equally long line segment \(P Q R T P\) on the pyramid's mantle.

Thus, the task is reduced to determining all \(\varphi\) with \(0^{\circ}<\varphi<90^{\circ}\) for which there exists a shortest among all the line segments mentioned in (1).

We investigate two cases:

1) \(0^{\circ}<4 \varphi<180^{\circ}\), i.e., the hexagon \(S^{\prime} A^{\prime} B^{\prime} C^{\prime} D^{\prime} A^{\prime \prime}\) is convex.

2) \(180^{\circ}<4 \varphi<360^{\circ}\), i.e., the hexagon \(S^{\prime}\) has an inward-pointing vertex or is degenerate into a pentagon.

Case 1: Due to the convexity of the hexagon, the line segment \(P^{\prime} P^{\prime \prime}\) intersects each of the segments \(S^{\prime} B^{\prime}, S^{\prime} C^{\prime}\), \(S^{\prime} D^{\prime}\) at exactly one internal point, which are denoted in this order as \(Q^{\prime}, R^{\prime}, T^{\prime}\).

Proof: Since the interior angle of the hexagon at \(S^{\prime}\) is \(4 \varphi\), \(P^{\prime}\) and \(P^{\prime \prime}\) lie on different sides of each of the lines through \(S^{\prime}\) and \(B^{\prime}\), through \(S^{\prime}\) and \(C^{\prime}\), and through \(S^{\prime}\) and \(D^{\prime}\). Therefore, \(P^{\prime} P^{\prime \prime}\) intersects each of these lines, and since the hexagon is convex, it intersects the segments \(S^{\prime} B^{\prime}, S^{\prime} C^{\prime}\), \(S^{\prime} D^{\prime}\) internally.

Thus, \(P^{\prime} Q^{\prime} R^{\prime} T^{\prime} P^{\prime \prime}\) is one of the line segments (1) and has the length \(P^{\prime} P^{\prime \prime}\), making it the shortest.

Case 2: In this case, either \(P^{\prime} P^{\prime \prime}\) contains the point \(S^{\prime}\) (if \(4 \varphi = 180^{\circ}\)) or \(P^{\prime} P^{\prime \prime}\) does not contain any point of the segments \(S^{\prime} B^{\prime}, S^{\prime} C^{\prime}, S^{\prime} D^{\prime}\).

If \(P^{\prime} Q^{\prime} R^{\prime} T^{\prime} P^{\prime \prime}\) is one of the admissible line segments (1), then for at least one of the three line segments \(P^{\prime} Q^{\prime} R^{\prime}, Q^{\prime} R^{\prime} T^{\prime}, R^{\prime} T^{\prime} P^{\prime \prime}\), not all three given points lie on the same line.

By replacing such a line segment with the line segment connecting its endpoints, the latter has a smaller length than the line segment and lies within the hexagon due to \(2 \varphi < 180^{\circ}\), forming another admissible line segment (1) of smaller length together with the remaining two segments of \(P^{\prime} Q^{\prime} R^{\prime} T^{\prime} P^{\prime \prime}\).

Therefore, there is no shortest among the line segments (1) in Case 2.

The set of angle measures \(\varphi\) sought is the set of all \(\varphi\) with \(0^{\circ}<\varphi<45^{\circ}\).

Adapted from \([5]\)