14. In a quiz, no two people had the same score and the score of each participant is equal to $n+2-2 k$ where $n$ is a constant and $k$ is the rank of the participant. If the total score of all participants is 2009, find the smallest possible value of $n$.
(2 marks)
In a quiz, no two people had the same score and the score of each participant is equal to $n+2-2 k$ where $n$ is a constant and $k$ is the rank of the participant. If the total score of all participants is 2009, find the smallest possible value of $n$.
14. 89
14. Let \( m \) be the number of people taking the quiz. Then we have
\[
\begin{aligned}
2009 & =n+(n-2)+(n-4)+\cdots+(n-2 m+2) \\
& =\frac{m}{2}[n+(n-2 m+2)] \\
& =m(n-m+1)
\end{aligned}
\]

Hence \( n=\frac{2009}{m}+m-1 \), where \( m \) is a factor of 2009. We check that \( m=41 \) gives the smallest value of \( n \), which is 89.