5. In the cash register of the commercial director Vasily, money appeared for the first time in a long time - 2019 coins. Unfortunately, it is known that one of them is counterfeit, differing in weight from the genuine ones. Enraged workers employed by Vasily are demanding immediate payment of their salaries, and only with genuine coins. The director has balance scales without weights. As soon as it becomes clear which coins are genuine, they are paid out to the next workers and, naturally, do not participate in further weighings. The curious Vasily wants to determine whether the counterfeit coin is lighter or heavier than the genuine ones. Can he definitely do this?
Solution. Answer: no. If each weighing results in equality, then all the coins used in it are genuine, and thus should be given away immediately. After this, a smaller odd number of coins will remain. Eventually, the fake coin will be the last one left, and Vasily will never learn its weight.