Question 1205113
To determine the minimum points that a soccer team must achieve to be ranked at least $ k $-th in a league with $ n $ teams, where each team plays every other team twice, we need to analyze the point distribution based on the match results.

### Step 1: Total Matches Played

In a league with $ n $ teams, each team plays every other team twice. The total number of matches played by each team is:

$$
\text{Total Matches} = 2 \times \binom{n}{2} = n(n-1)
$$

### Step 2: Points Distribution

Each match awards points based on the result:
- Win: $ a $ points
- Draw: $ b $ points
- Loss: $ c $ points

Given the conditions:
- $ a + c > 2b $ (this ensures that winning is more rewarding than drawing)
- $ b > c $ (this ensures that drawing is better than losing)

### Step 3: Maximum Points Possible

The maximum points a team can earn is if they win all their matches. The total number of matches for each team is $ 2(n-1) $ (since each team plays $ n-1 $ other teams twice). Therefore, the maximum points a team can earn is:

$$
\text{Max Points} = 2(n-1) \times a
$$

### Step 4: Minimum Points Required to be at Least $ k $-th

To find the minimum points required to be ranked at least $ k $-th, we need to consider the distribution of points among the teams. 

1. **Total Points in the League**: The total points distributed in the league can be calculated as follows:
   - Each match results in a total of $ a $ points (if there is a win) or $ 2b $ points (if there is a draw).
   - The total number of matches is $ n(n-1) $, so the total points distributed in the league is:
   $$
   \text{Total Points} = n(n-1) \times \text{Average Points per Match}
   $$

2. **Points Required for $ k $-th Place**: To ensure that a team is at least $ k $-th, we need to consider the points of the top $ k-1 $ teams. The minimum points required can be estimated as follows:
   - If we assume that the top $ k-1 $ teams have the maximum possible points, the team in $ k $-th place must have more points than the $ k $-th team.
   - Therefore, the minimum points required can be approximated as:
   $$
   \text{Minimum Points} = \left\lceil \frac{\text{Total Points}}{n} \right\rceil + 1
   $$

### Conclusion

The minimum points that the soccer team must achieve to be ranked at least $ k $-th in the final ranking can be summarized as:

$$
\text{Minimum Points} = \left\lceil \frac{(n(n-1) \times \text{Average Points per Match})}{n} \right\rceil + 1
$$

Where:
- $ a $ is the points for a win,
- $ b $ is the points for a draw,
- $ c $ is the points for a loss,
- $ n $ is the number of teams,
- $ k $ is the desired rank.

This formula provides a general guideline for determining the minimum points needed based on the league's structure and point distribution rules.