작은 선분 길이의 합

$\displaystyle \mathrm{\overline{P _{\mathit{n}} \mathrm{Q} _{\mathit{n}}}} = \mathit{n} - \frac{1}{20} n \left( n + \frac{1}{3} \right)$, $\displaystyle \mathrm{\overline{Q _{\mathit{n}} \mathrm{R} _{\mathit{n}}}} = \frac{1}{20} n \left( n + \frac{1}{3} \right)$이다. $\displaystyle \mathrm{\overline{P _{\mathit{n}} \mathrm{Q} _{\mathit{n}}}} > \mathrm{\overline{Q _{\mathit{n}} \mathrm{R} _{\mathit{n}}}}$을 만족하는 $n$의 범위를 구하면 $\displaystyle n - \frac{1}{20} n \left( n + \frac{1}{3} \right) > \frac{1}{20} n \left( n + \frac{1}{3} \right)$ $\displaystyle n > \frac{1}{10} n \left( n + \frac{1}{3} \right)$ $n$은 자연수이므로 $\displaystyle 1 > \frac{1}{10} \left( n + \frac{1}{3} \right)$, $\displaystyle n < 10 - \frac{1}{3}$ $\therefore$$\displaystyle a _{n} = \begin{cases} \mathrm{\overline{Q _{\mathit{n}} \mathrm{R} _{\mathit{n}}}} = \frac{1}{20} n \left( n + \frac{1}{3} \right) & \left( 1 \leq n \leq 9 \right) \\ \mathrm{\overline{P _{\mathit{n}} \mathrm{Q} _{\mathit{n}}}} = n - \frac{1}{20} n \left( n + \frac{1}{3} \right) & \left( n \geq 10 \right) \end{cases}$ $\therefore$$\displaystyle \sum\limits _{n=1} ^{10} a _{n}$$\displaystyle = \sum\limits _{n=1} ^{9} \frac{1}{20} n \left( n + \frac{1}{3} \right) + a _{10}$ $\displaystyle = \frac{1}{20} \times \frac{9 \times 10 \times 19}{6} + \frac{1}{60} \times \frac{9 \times 10}{2}$$\displaystyle + 10 - \frac{1}{20} \times 10 \left( 10 + \frac{1}{3} \right)$ $\displaystyle = \frac{119}{6}$