54. There are 100 identical-looking coins. It is known that exactly 4 of them are counterfeit and weigh the same, but are lighter than the genuine ones. How can you find at least 13 genuine coins in 2 weighings using a balance scale without weights?
Solution. Place 29 coins on each pan. In the heavier group, there is no more than one fake coin. Place 14 coins from it on each pan, in the heavier one - all are genuine, in case of equality: all 28 are genuine. In case of equality the first time, among the remaining 42 coins there is an even number of fake ones. Add 13 coins from the other to one pan and compare with 42 coins. Now there are no less than 2 fakes on the scales. In case of equality, there are 2 fakes on each, so the remaining 16 are genuine. If the 42 coins are lighter, then there are either 4 fakes or two. In both cases, the relocated 13 are genuine. If the 42 coins are heavier, then these are 42 genuine ones.