A tournament will take place with 100 competitors, all with different skill levels. The most skilled competitor always wins against the least skilled competitor. Each participant plays exactly twice, with two randomly drawn opponents (once against each). A competitor who wins two matches receives a medal. Determine the minimum number of medals that can be distributed in the tournament.
Solution

Let's denote that a player $A$ is more skilled than a player $B$ by $A \rightarrow B$. Suppose that

$$
L_{1} \rightarrow L_{2} \rightarrow L_{3} \rightarrow \ldots \rightarrow L_{100}
$$

are the 100 players in the tournament. Since $L_{1}$ is the most skilled of all, he will certainly win a medal by winning his two matches. We will show that this is the minimum number of medals by providing an example of a tournament where he is the only one to win it. In the first round of the tournament, consider the matches:

$$
L_{100} \times L_{99}, L_{98} \times L_{97}, \ldots, L_{2} \times L_{1}
$$

Thus, all players with odd numbers have lost at least one match. In the second round, consider the matches:

$$
L_{100} \times L_{1}, L_{99} \times L_{98}, \ldots, L_{3} \times L_{2}
$$

Thus, all players with even numbers have lost at least one match. In this situation, only $L_{100}$ will win a medal.