Example 5 In a certain competition, each player plays exactly one game against each of the other players. A win earns the player 1 point, a loss 0 points, and a draw 0.5 points for each player. After the competition, it is found that exactly half of each player's points were earned in games against the 10 lowest-scoring players (the 10 lowest-scoring players earned half of their points in games against each other). Find the total number of players in the competition.
(3rd American Invitational Mathematics Examination)
Solution: Let there be a total of $n$ players, then they collectively score $\mathrm{C}_{n}^{2}$ points. The 10 players with the lowest scores collectively score $\mathrm{C}_{10}^{2}=45$ points through their matches against each other, which is half of their total score, so these 10 players collectively score 90 points. The remaining $n-10$ players collectively score $\mathrm{C}_{n-10}^{2}$ points through their matches against each other, which is also half of their total score, so these $n-10$ players collectively score $2 \mathrm{C}_{n-10}^{2}$ points. Therefore, we have
$$
\mathrm{C}_{n}^{2}=90+2 \mathrm{C}_{n-10}^{2} \text {, }
$$

which simplifies to $n^{2}-41 n+400=0$.
Solving this, we get $n=16$ or 25.
However, if there are only 16 players, then there would be 6 high-scoring players, who collectively score 30 points, averaging 5 points per person. Meanwhile, the 10 low-scoring players average $90 \div 10=9$ points per person, which is a contradiction.
Therefore, $n=25$, meaning there are a total of 25 participants.