LV OM - III - Task 5

Determine the maximum number of lines in space passing through a fixed point and such that any two intersect at the same angle.
Let $ \ell_1,\ldots,\ell_n $ be lines passing through a common point $ O $. A pair of intersecting lines determines four angles on the plane containing them: two vertical angles with measure $ \alpha \leq 90^\circ $ and the other two angles with measure $ 180^\circ - \alpha $. According to the assumption, the value of $ \alpha $ is the same for every pair $ \ell_i, \ell_j $.
Consider a sphere $ S $ centered at $ O $ with an arbitrary radius, and denote by $ A_i $, $ B_i $ the points of intersection of the line $ \ell_i $ with this sphere. Each of the segments $ A_iA_j $, $ A_iB_j $, $ B_iB_j $ (for $ i \neq j $) is a chord of the sphere $ S $, determined by a central angle of measure $ \alpha $ or $ 180^\circ - \alpha $. Therefore, these segments have at most two different lengths $ a $ and $ b $.
Fix the notation so that $ \measuredangle A_iOA_n = \alpha $ for $ i = 1,\ldots,n-1 $. Then the points $ A_1, \ldots, A_{n-1} $ lie on a single circle (lying in a plane perpendicular to the line $ \ell_n $). Let $ A_1C $ be the diameter of this circle. Each of the points $ A_2,\ldots,A_{n-1} $ is at a distance $ a $ or $ b $ from the point $ A_1 $; hence, on each of the two semicircles with endpoints $ A_1 $ and $ C $, there are at most two points from the set $ \{A_2,\ldots ,A_{n-1}\} $. Therefore, this set has at most four elements; which means that $ n \leq 6 $.
On the other hand, if $ n = 6 $, we can place the points $ A_1,A_2,\ldots,A_5 $ at the vertices of a regular pentagon, and the plane of this pentagon at such a distance from the point $ A_6 $ that these six points, together with their antipodal points $ B_1,\ldots,B_6 $, are the vertices of a regular icosahedron. The segments $ A_iB_i $ (diameters of the sphere $ S $) connect opposite vertices of this icosahedron and any two of them form the same angle. Therefore, $ n = 6 $ is the largest possible number of lines $ \ell_i $.