Berdnikov A.

On a spherical planet with an equator length of 1, they plan to lay $N$ circular roads, each of which will run along a circle of length 1. Then, several trains will be launched on each road. All trains will travel along the roads at the same positive constant speed, never stopping or colliding. What is the maximum possible total length of all the trains under these conditions? Consider the trains as arcs of zero thickness, with the endpoints removed. Solve the problem in the cases: a) $N=3$; b) $N=4$.
Evaluation. Let's take any two roads - great circles on a sphere. They intersect at some point-node. Mentally rotate one of these roads relative to the diameter containing the node to align the roads and the directions of movement on them. If in this experiment the arc-trains intersect, then after some time they will actually intersect at the node, which is prohibited. Therefore, the sum of the lengths of the trains on these roads is no more than 1. Let $a_{1}, \ldots, a_{n}$ be the sums of the lengths of the trains on $n$ roads. By summing all inequalities of the form $a_{i}+a_{j} \leq 1$, where $1 \leq i<j \leq n$, we get

$(n-1)\left(a_{1}+\ldots+a_{n}\right) \leq 1 / 2 n(n-1)$, that is, $a_{1}+\ldots+a_{n} \leq n / 2$.

## Example. First method.

![](https://cdn.mathpix.com/cropped/2024_05_06_7dfbf12cfa0901bdf2bfg-24.jpg?height=389&width=1012&top_left_y=0&top_left_x=526)

In the figures, "projections" of road plans are shown on inscribed in the sphere: a) octahedron (left); b) cuboctahedron (right). Each road is highlighted in its own color. Edges with arrows are trains.

(The cuboctahedron is obtained from a cube by connecting the midpoints of all adjacent edges. Thus, the projections of the roads are regular hexagons - sections of the former cube.)

Second method. b) Consider a regular quadrilateral prism inscribed in the given sphere. Each road is a section of the sphere by a plane passing through one of the edges of the base and the opposite edge of the other base of the prism. Place one train of length $1 / 2$ on each road. In the figure, "projections" of the roads and trains on the surface of the prism are shown (each road corresponds to its own color).

a) It is sufficient to remove one of the roads constructed in part b.

![](https://cdn.mathpix.com/cropped/2024_05_06_7dfbf12cfa0901bdf2bfg-24.jpg?height=286&width=328&top_left_y=820&top_left_x=865)

## Answer

a) 1.5 ; b) 2.