2000 IMO Shortlist S07

Let $a, b, c$ be positive integers satisfying the conditions $b > 2a$ and $c > 2b.$ Show that there exists a real number $\lambda$ with the property that all the three numbers $\lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $\left(\frac {1}{3}, \frac {2}{3} \right].$

设 $a, b, c$ 为正整数，满足条件 $b > 2a$ 且 $c > 2b.$ 证明存在实数 $\lambda$，且三个数 $\lambda a, \lambda b, \lambda c$ 的小数部分均位于区间 $\left(\frac {1}{3}, \frac {2}{3} \right].$