Let's tackle both of your problems one by one.
### Problem 19
We begin by noting a crucial detail about the total interactions between the teams. There are six teams, and each team plays every other team twice:
1. Total games played by one team:
Each team plays each of the other 5 teams twice. Therefore, each team plays:
[tex]\[
5 \times 2 = 10 \text{ games.}
\][/tex]
2. Calculating the maximum possible results:
There are certain combinations of results (wins, draws, losses) that could yield the highest scores for three teams, which end up tied.
Let's assume the scenario where the top three teams win most of their matches against the bottom three teams and only draw among themselves.
3. Estimating the highest score:
- Suppose the top teams win all their matches against the bottom three teams (6 matches each). This yields:
[tex]\[
6 \text{ wins} \times 3 \text{ points per win} = 18 \text{ points.}
\][/tex]
- They play 4 matches among themselves (since they play each other twice). If these 4 matches are all draws:
[tex]\[
4 \text{ draws} \times 1 \text{ point per draw} = 4 \text{ points.}
\][/tex]
Adding these scores together gives:
[tex]\[
18 + 4 = 22 \text{ points.}
\][/tex]
So, the greatest possible total points for each of the top three teams is:
Answer: (A) 22
### Problem 20
Given in the problem, we have:
- Initial number of computers: 9
- Additional computers added every day: 5
- Number of installation days: 4
To find the total number of computers, we break down the steps:
1. Calculate the number of added computers:
- Additional computers added per day:
[tex]\[
5 \text{ computers.}
\][/tex]
- For 4 days:
[tex]\[
5 \text{ computers/day} \times 4 \text{ days} = 20 \text{ computers.}
\][/tex]
2. Calculate the total number of computers:
- Combine the initial number of computers and the added computers:
[tex]\[
9 \text{ initial computers} + 20 \text{ added computers} = 29 \text{ computers.}
\][/tex]
Thus, 20 is the number of computers added, and 29 is the total number of computers after 4 days.
Answer:
- Number of computers added: 20
- Total number of computers: 29
