1997 Putnam B1

Let $\{x\}$ denote the distance between the real number $x$ and the

nearest integer. For each positive integer $n$, evaluate

$$F_n=\sum_{m=1}^{6n-1} \min(\{\frac{m}{6n}\},\{\frac{m}{3n}\}).$$

(Here $\min(a,b)$ denotes the minimum of $a$ and $b$.)

令 $\{x\}$ 表示实数 $x$ 与实数 $x$ 之间的距离

最接近的整数。对于每个正整数 $n$，计算

$$F_n=\sum_{m=1}^{6n-1} \min(\{\frac{m}{6n}\},\{\frac{m}{3n}\}).$$

(这里$\min(a,b)$表示$a$和$b$中的最小值。)