ZOJ Problem Set - 2759
Michael is interested in physics. During his playing with balances and counterpoises, he finds that if he chooses a set of suitable counterpoises, he can weigh many more different weights than others. After a series of hard work, Michael decides to choose the counterpoises which have the weight 1,3,9,27,....,exactly all the power exponents of 3. He thinks he can weigh every weight with these counterpoises. But Alex is doubt about this. He says he can give an object which cannot be weighed by this set of counterpoises. Now Michael is asking for your help to win the bet. He also tells you a secret to help: "The counterpoises could be placed on both sides of the balance. As well, the weight of all Alex's objects are no more than integers. Cheer up! You'll win."
Author: JI, Cheng
Source: ZOJ Monthly, September 2006