[Ribalko A.V.

In the National Basketball Association, there are 30 teams, each of which plays 82 games with other teams in the regular season. Can the Association's management divide the teams (not necessarily equally) into Eastern and Western Conferences and create a schedule such that the games between teams from different conferences make up exactly half of the total number of games?

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Let $x$ and $y$ be the total number of matches played within the Eastern and Western conferences, respectively, and $z$ be the number of matches between teams from different conferences. We need to prove that the equation $z = 1/2(x + y + z)$ is impossible.

Each of the $k$ teams in the Eastern conference plays 82 games; thus, $82k = 2x + z$ (the factor 2 appears because each internal match is counted for both participating teams). From this, it is clear that $z$ is even. However, from the calculation of the total number of matches $x + y + z = 30 \cdot 82 / 2$, it follows that the number $1/2(x + y + z) = 15 \cdot 41$ is odd. Therefore, the equation $z = 1/2(x + y + z)$ cannot be satisfied.

## Answer

It cannot.

Author: Galperin G.A.

There is a convex polyhedron with 100 edges. All its vertices are cut off by planes close to the vertices (i.e., so that the planes do not intersect each other inside or on the boundary of the polyhedron). Find the number of

a) vertices;

b) edges.

## Solution

a) On each edge of the original polyhedron, there are two vertices of the resulting polyhedron. Therefore, the resulting polyhedron has $2 \cdot 100 = 200$ vertices.

b) From each vertex of the new polyhedron, three edges emanate. Thus, the resulting polyhedron has $200 \cdot 3 / 2 = 300$ edges.

## Answer

200 vertices and 300 edges.