What is the maximum number of points we can mark in space so that the line connecting any two of them is tangent to a given sphere and no three of them lie on the same line?
The answer to the question is 4. The inscribed sphere of the cube shown in Figure 1 touches every edge of the regular tetrahedron $ABCD$, because the midpoints of the edges are precisely the centers of the cube's faces, and the inscribed sphere touches the faces exactly at their centers.

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Figure 1

We will show that 5 points cannot be placed in accordance with the conditions. Suppose that the line connecting any two of the points $A, B, C, D, E$ touches the sphere $g$. Tangents drawn from an external point to a sphere form a cone that touches the sphere in a circle. Thus, to each of the points $A, B, C, D, E$, a circle $a, b, c, d, e$ can be associated on $g$. We will show that these circles touch each other pairwise.

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For example, if the line $AB$ touches $g$ at point $P$, then $P$ lies on both circles $a$ and $b$ (Figure 2). If there were another point $Q$ different from $P$ on both circles, then $AQ$ and $BQ$ would also be tangents, and $AP = AQ$ and $BP = BQ$ would hold, which contradicts the triangle inequality $AP + BP = AB < AQ + QB$. Therefore, any two of the circles $a, b, c, d, e$ on the sphere $g$ touch each other. Since the points of tangency of the circles lie on the line connecting the corresponding points, and any three of the points $A, B, C, D, E$ are not collinear, the points of tangency of the circles are all distinct.

![](https://cdn.mathpix.com/cropped/2024_05_02_ae9a36517bac5902e057g-1.jpg?height=426&width=420&top_left_y=1874&top_left_x=836)

Let the points of tangency of circle $a$ with the other four circles be denoted by $E_b, E_c, E_d,$ and $E_e$. We can assume that points $E_b$ and $E_c$ separate points $E_d$ and $E_e$ on the circle (Figure 3). Two of the circles $b, c, d,$ and $e$ can only touch if they lie in the same half-space determined by the plane of circle $a$. Therefore, all four circles lie in the same half-space. However, the plane determined by the points of tangency of circles $b$ and $c$, and points $E_b$ and $E_c$, separates circles $d$ and $e$, so they do not touch. This contradiction shows that 5 points cannot be placed in accordance with the conditions.