1. ## Football game

15 football teams competed in a tournament. Each team played against all the other teams exactly once. A win gave 2 points, a draw 1 point and a loss 0 points. At the end of the tournament every team had different set of points. The team with the least points got 7 points. Prove that the winning team had at least one draw.

2. The total number of matches played is 14 + 13 + 12 +...+1 = 105 matches. Two points are awarded for each match, so in the tournament the total number of points awarded is 2*105 = 210 points. If the last place team got 7 points, and no two teams got the same number of points, then the point totals for each team from lowest to highest must be:

7, 8, 9, 10, ..., 20, 21

This is the only set of results that works out properly. Since the wining team got 21 points, and the only way to get an odd number of points is to have an odd number of ties. So the winning team must have had at least 1 tie.

3. then the point totals for each team from lowest to highest must be:

7, 8, 9, 10, ..., 20, 21

This is the only set of results that works out properly. Since the wining team got 21 points
This is false. First, you didn't prove anything by saying "this is the only set of results that works out properly". Second, there are 15 matches for the first team so he could got 29 points. Thus this is not the only set of results that works.
To solve the problem, I believe, you must suppose that the first team won all his matches. So that he got 30 points, and then shows it leads to a contradiction.

4. ## Im sorry...

Originally Posted by arbolis
This is false. First, you didn't prove anything by saying "this is the only set of results that works out properly". Second, there are 15 matches for the first team so he could got 29 points. Thus this is not the only set of results that works.
To solve the problem, I believe, you must suppose that the first team won all his matches. So that he got 30 points, and then shows it leads to a contradiction.
Actually the first team does not play fifteen games. In that case it would have to play against itself. Think about it, there are fourteen other teams that it will play against...fourteen. This means fourteen games. And the second team will also play against fourteen teams (minus the game against the first team that was already played) which equals thirteen. This goes on and on. 14+13+12.....+1!

And also. If the winning team got full points, theres no way for the points to be distributed so that the last team got 7 points and the other teams something in between (and satisfying the condition that no team has the same amount of points)

Think about it. Have a nice day!

5. Actually the first team does not play fifteen games. In that case it would have to play against itself.
Sorry you're right. I thought about it, but then forgot it when I wrote the post!
And also. If the winning team got full points, theres no way for the points to be distributed so that the last team got 7 points and the other teams something in between (and satisfying the condition that no team has the same amount of points)
This is what we must prove. It's not so evident. It is true, but how to show it? Suppose the first team won all his matches, so 14 matches. He got 28 points.
The first got 28 points, the last 7. How can you show that the 13 other teams cannot have different points between 7 and 28? It remains a range of 20 possible scores for 13 teams. Of course if you try you'll find out that it's impossible, but it's not rigorous.
There are $210$ points to give. $210-35=75$ points to give to 13 teams.
Say you give 8 points to a team, 9 to another, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. It makes 182 points. Added to the 35 given points, it make 217 which is not 210. So we see that the first team cannot have won all his games (it doesn't mean it has at least one draw). As the score got overpassed by 7 points, the problem is not finished and not evident. It's more complicated.
I repeat, from
And also. If the winning team got full points, theres no way for the points to be distributed so that the last team got 7 points and the other teams something in between (and satisfying the condition that no team has the same amount of points)
, you must show why there's no way to distribute the points as it must be.
Have a nice day you too.

6. Originally Posted by arbolis
Sorry you're right. I thought about it, but then forgot it when I wrote the post!
And about This is what we must prove. It's not so evident. It is true, but how to show it? Suppose the first team won all his matches, so 14 matches. He got 28 points.
The first got 28 points, the last 7. How can you show that the 13 other teams cannot have different points between 7 and 28? It remains a range of 20 possible scores for 13 teams. Of course if you try you'll find out that it's impossible, but it's not rigorous.
There are $210$ points to give. $210-35=75$ points to give to 13 teams.
Say you give 8 points to a team, 9 to another, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. It makes 182 points. Added to the 35 given points, it make 217 which is not 210. So we see that the first team cannot have won all his games (it doesn't mean it has at least one draw). As the score got overpassed by 7 points, the problem is not finished and not evident. It's more complicated.
I repeat, from , you must show why there's no way to distribute the points as it must be.
Have a nice day you too.

Ok....let me try a different approach. Your totally right about the proving perspective. We should strive to a more mathematical proving point. But we don´t really need one. First off don´t assume anything with the winning team, you mess it only up for yourself. Let´s start with the last team that got 7 points. Now which is least amount of points that could be given in the tournament given the conditions. Well since there can only be given whole numbers as points it should be 7,8,9,10...21 because a team cannot get 13,7 points. You get it? The last team gets 7 points so the second last team should get at least 8 or more because it cannot get 7 and 8 is ok. Remember were trying to get the least amount of points just for trying. So the third get 9 and so on. Now fifteen teams (now assuming)gets points from 7,8,9...to 21. Sum these up and you get 210. A different distribution of points would end up either in having a bigger total (which is not allowed) or in a few teams having the same points (which is not either allowed).

And since the winning team has 21 points it must have played at least one draw.

7. Originally Posted by lynch-mob
Let´s start with the last team that got 7 points. Now which is least amount of points that could be given in the tournament given the conditions. Well since there can only be given whole numbers as points it should be 7,8,9,10...21 because a team cannot get 13,7 points. You get it?
I am finding it hard to understand. Why can the winning team score more than 21 points?

8. Originally Posted by lynch-mob
15 football teams competed in a tournament. Each team played against all the other teams exactly once. A win gave 2 points, a draw 1 point and a loss 0 points. At the end of the tournament every team had different set of points. The team with the least points got 7 points. Prove that the winning team had at least one draw.
The teams play a total of C(15,2) = 105 games and each game generates a total of 2 points for the two teams involved, so the total score of all the teams is 2 * 105 = 210.

On the other hand, the team with the least points got 7 points, so the total score of all the teams is at least 7 + 8 + ... + 21 = 210 points, and this value is obtained only when the scores are 7, 8, .., 21; any other possibility generates a total which is strictly greater than 210. Since we know the total is 210, the scores must be the successive integers 7 to 21, i.e., the winning team scored 21 points.

Since 21 is odd the winner must have scored at least one draw; otherwise the total would be a sum of 0s and 2s, and even.

9. and this value is obtained only when the scores are 7, 8, .., 21; any other possibility generates a total which is strictly greater than 210.
You did it! I was close, but you finished it. This is well proved. Bravo!!!

10. Didn't anyone notice that the very first comment above, by ebaines, gave a complete answer to this problem (based on the simple observation that the 15 consecutive integers starting from 7 already have a sum of 210, and must therefore be the scores of the 15 teams)?

11. Didn't anyone notice that the very first comment above, by ebaines, gave a complete answer to this problem (based on the simple observation that the 15 consecutive integers starting from 7 already have a sum of 210, and must therefore be the scores of the 15 teams)?
For me it wasn't all done. Of course, what he said is true. But he didn't explain
This is the only set of results that works out properly.
while awkward said
any other possibility generates a total which is strictly greater than 210.
which in my opinion complete the proof.

12. ## ?

Originally Posted by Isomorphism
I am finding it hard to understand. Why can the winning team score more than 21 points?
Who said it can get more than 21 points?