Question

Five chess players took part in the tournament. It is known that each chess player played the same game with the others, and:
A) the leader did not make a single draw;
B) runner-up did not lose a single game;
C) who won the fourth place did not win a single game.
For the victory is awarded one point, for a draw ½ points. Determine the results of all games.

Answer 1

From the fact that it is a question of the leader, of the runner-up, of the fourth place among the five participants, it follows that they all scored a different number of points, no place was divided between the participants in the tournament.

The leader did not make a single draw - it means he has 4 defeats, or 1 win and 3 losses, or 2 wins and 2 losses, or 3 wins and 1 loss, or 4 wins, (respectively 0, 1, 2, 3 or 4 points )

Since the runner-up did not lose a single game, i.e. (Either 2 draws), or 3 draws 1 win (2.5 points), or 2 draws 2 wins (3 points), or 1 draw 3 wins (3.5 points), or 4 wins ( 4 points), then the second minimum 2 points

So the leader (obviously scored more than the runner-up, otherwise they would have shared the lead), scored either 3 points or 4 points.

But since the second did not lose to the first, and they did not play a draw (since the first did not play a single draw, and the draw for the first one - is equal to a draw for the second in their game), the leader lost the game to the second and his result 3 points (3 wins 1 defeat (second in a row))

So the second 1 victory (the first) and he scored less than 3 points, so he scored 2.5 points (a victory over the first, 3 draws with the rest)

Next, the fourth player did not win a single game, i.e. (4 points), 3 losses and 1 draw (0.5 points), 2 losses and 2 draws (1 point), 1 loss and 3 draws (1.5 points), 4 draws (2 points) )Option 4 losses is impossible, because the fourth place should bring more points than the fifth.Since the fourth loss to the first option is 4 draws impossible

If the fourth played 3 losses 1 draw (0.5 eyes), then the fifth lost all games (0 eyes), but this is impossible because the fifth played with the second in a draw.

If the fourth played 2 losses and 2 draws (1 point), then given that the fifth 0 eyes can not be due to a draw with the second, and should be less than the fourth then the fifth 0.5 eyes, which means thatThe fifth lost to the first, a draw with the second, and hence the rest - with the third and fourth lost, but the fourth could not win from the fifth, so the option is impossible

If the fourth played 1 loss and 3 draws (1.5 points), then it means that the second has 2 points (more than 1.5 for the fourth less than 2.5 for the second)Then the third losing the game to the first, a draw with the second and fourth (1 point) was to win at the fifthThen the fifth defeat to the first, draw with the second, draw with the fourth, and defeat to the third (1 point)

In total we obtain:
♦ LEADER (loss to the second, with the third, fourth, fifth - victory) - 3 points
♦ SECOND PLACE (first, third, fourth, fifth-not) -2.5 points
♦ THIRD PLACE (losing at the first, winning the fifth, draw with the second and fourth) - 2 points
♦ THE FOURTH PLACE (the loss to the first, with the second, third and fifth - a draw) - 1.5 points
♦ FIFTH PLACE (defeat to the first and third, draw with the second and fourth) - 1 point