세 자연수 역수합의 최솟값

$x < y < z$라 하면 $\displaystyle 1 < \frac{1}{x} + \frac{1}{y} + \frac{1}{z} < \frac{3}{x}$ $\rightarrow$ $\displaystyle 1 < \frac{3}{x}$ $\rightarrow$ $x < 3$ $x , y , z$가 $1$보다 큰 자연수이므로 $\therefore x = 2$ 또한, $\displaystyle \frac{1}{2} + \frac{1}{y} + \frac{1}{z} > 1$, $\displaystyle \frac{1}{2} < \frac{1}{y} + \frac{1}{z} < \frac{2}{y}$ $\rightarrow$ $\displaystyle \frac{2}{y} > \frac{1}{2}$$\rightarrow$ $y < 4$ $x < y$이므로 $\therefore y = 3$ 또한, $\displaystyle \frac{1}{2} + \frac{1}{3} + \frac{1}{z} > 1$$\rightarrow$ $\displaystyle \frac{1}{z} > \frac{1}{6}$ $\rightarrow$$z < 6$ $y < z$이므로 $\therefore z = 4 or 5$ 따라서, $\displaystyle \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$의 최솟값은 $\displaystyle \frac{1}{2} + \frac{1}{3} + \frac{1}{5} = \frac{31}{30}$