8.4. In a chess tournament, each of the 10 players played one game against each other, and Petya came in last place (scored fewer points than any other participant). Then two players were disqualified, and all points earned in matches against them were annulled, and these two players were removed from the table. It turned out that as a result, Petya became the winner of the tournament (scored more points than any other participant). How many points did Petya end up with (after the disqualification of the players)? 1 point is awarded for a win, 0.5 points for a draw, and 0 points for a loss.
Answer: 4 points.

Solution. In a tournament with 10 players, played in a single round, $\frac{10 \cdot 9}{2}=45$ points are distributed. Therefore, there will be a player who scores no more than $45: 10=4.5$ points. This means that the player who finished in absolute last place scored no more than 4 points. Similarly, in a tournament with $10-2=8$ players, played in a single round, $\frac{8 \cdot 7}{2}=28$ points are distributed. In such a tournament, there will be a player who scores no less than $28: 8=3.5$ points. This means that the player who finished in absolute first place (after the disqualification) scored no less than 4 points. Therefore, Petya could have scored only 4 points.

Comment. A correct answer without justification - 0 points.

A correct answer obtained by considering an example - 1 point.

One of the two estimates for the number of points Petya scored (before or after disqualification) is correctly proven - 3 points.

Note. The tournament described in the problem is possible. Providing an example of such a tournament is not required, as it follows from the problem statement that such a tournament exists. Points are not deducted for the lack of such an example.