7. A row of 100 coins is laid out. All the coins look the same, but somewhere among them are 50 counterfeit ones (and the rest are genuine). All genuine coins weigh the same, and the counterfeit ones may weigh differently, but each counterfeit is lighter than a genuine one. Can at least 34 genuine coins be found with a single weighing on a balance scale without weights? (O. Dmitriev, R. Zhenodarov)
Answer. It is possible. Solution. Number the coins from left to right with numbers from 1 to 100. Compare coins 17 and 84. At least one of them is genuine. Therefore, if the scales are in balance, both coins are genuine; in this case, 34 coins with numbers 1-17 and 84-100 will be genuine, since 50 counterfeit coins do not fit in these intervals. Now let's assume that coin 17 outweighs. Then it is genuine, and coin 84 is counterfeit. Since the numbers of any two counterfeit coins differ by no more than 49, in this case, the smallest number of a counterfeit coin is no less than $84-49=35$, that is, coins $1-34$ are definitely genuine. If coin 84 outweighs, similar reasoning shows that coins 67-100 are genuine.