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On a circle, points $A_{1}, A_{2}, \ldots, A_{16}$ are given. We will construct all possible convex polygons whose vertices are among the points $A_{1}, A_{2}, \ldots, A_{16}$. We will divide these polygons into two groups. The first group will include all polygons where $A_{1}$ is a vertex. The second group will include all polygons where $A_{1}$ is not a vertex. In which group are there more polygons?
To each polygon that does not have a vertex $A_{1}$, we associate a polygon with the vertex $A_{1}$ by simply adding $A_{1}$ to its vertices. The reverse operation (removing the vertex $A_{1}$) is impossible for triangles.

## Answer

In the first.

Five people play several domino games (two on two) such that each player has each of the others as a partner once and as an opponent twice. Find the number of games played and all ways of distributing the players.

## Solution

We can assume that the 1st and 2nd played against the 3rd and 4th. Then, in the game where the 1st was a partner of the 5th, the 2nd and one of the pair $\{3,4\}$ (let's say the 3rd) necessarily played against them.

The 3rd player has already been an opponent of the 1st twice, so the 2nd and 5th play against the 1st and 4th. After this, the 2nd cannot play against the 1st again, so the 4th and 5th play against the 1st and 3rd. The last game remains: the 2nd and 4th play against the 3rd and 5th.

## Answer

5 games; the distribution of players is unique (up to numbering).