Example) There are 10 people, and 2 people are playing a 1v1 game. Each player has to play once other, and if they win, they get 1 point. There are no ties.




Problem 1) What is the total number of games played by the ten players?




Problem 2) There is at most one person with a score of 9 points. Why is that?




Problem 3) What is the maximum number of people with a score of 8 or higher?


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문제 3번은 o1, gemini 띵킹은 풀 수 있음. 답은 3임.


근데 experimental-router-0112은 완전히 틀림













Sure, let's tackle each problem step by step.


---


**Problem 1) What is the total number of games played by the ten players?**


**Solution:**


Each of the 10 players must play against every other player exactly once. To determine the total number of unique games (or matches) that will be played, we can use the combination formula:


\[

\text{Number of games} = \binom{10}{2} = \frac{10 \times 9}{2} = 45

\]


**Answer:** **45 games** are played in total.


---


**Problem 2) There is at most one person with a score of 9 points. Why is that?**


**Explanation:**


In this setup:


- **Total Players:** 10

- **Each Player's Total Games:** 9 (since they play against each of the other 9 players)

- **Points Per Win:** 1

- **No Ties Allowed**


If a player scores **9 points**, it means they have won all their 9 matches. 


- **Implication:** Each of the other 9 players must have lost at least one game (specifically, to this undefeated player).

  

Therefore, it's **impossible** for more than one player to achieve a 9-point score because:


- If **Player A** has 9 points, then every other player has at least one loss (to **Player A**).

- If we try to have **Player B** also with 9 points, **Player A** and **Player B** would both need to have beaten each other, which is impossible since each game results in only one winner.


**Answer:** **Because if one player wins all 9 of their games, every other player must have at least one loss. Thus, only one person can achieve a perfect score of 9 points.**


---


**Problem 3) What is the maximum number of people with a score of 8 or higher?**


**Solution:**


Let's determine the maximum number of players that can each have **8 or more points**.


1. **Total Number of Games:** 45

2. **Total Points Distributed:** 45 (since each game awards exactly 1 point to the winner)

3. **Let’s assume \( k \) players have at least 8 points each.**

   

   - **Minimum total points for these \( k \) players:** \( 8k \)

   - **Remaining points for the other \( 10 - k \) players:** \( 45 - 8k \)


4. **Since no player can score more than 9 points, we also need to ensure that \( 8k \) does not exceed 45:**


   \[

   8k \leq 45 \implies k \leq \frac{45}{8} \approx 5.625

   \]

   

   Since \( k \) must be an integer, the maximum possible \( k \) is **5**.


5. **Verification:**

   

   - **5 players** scoring **8 points** each: \( 5 \times 8 = 40 \) points

   - **Remaining 5 players** would share **5 points** (each could have 1 point)

   - This distribution is possible without any contradictions.


**Answer:** **A maximum of five players** can have a score of **8 or higher**.