8.5. Petya has 4 Soviet copper coins - one each of 1, 2, 3, and 5 kopecks. He learned this fact: these coins should weigh exactly as many grams as their denomination. Petya wants to check this fact using a balance scale. Will he be able to do this if he has only one 1-gram weight?
8.5. Petya has 4 Soviet copper coins - one each of 1, 2, 3, and 5 kopecks. He learned this fact: these coins should weigh exactly as many grams as their denomination. Petya wants to check this fact using a balance scale. Will he be able to do this if he has only one 9-gram weight?

Answer: Yes, he can. Hint: Let \( x \) be the weight of the 1-kopeck coin, and \( y \) be the weight of the 2-kopeck coin. If the fact being checked is true, then \( (x + y) \) is the weight of the 3-kopeck coin, and \( (x + 2y) \) is the weight of the 5-kopeck coin. To verify this, we will perform two weighings: 1) 1 kopeck + 2 kopecks = 3 kopecks (?); 2) 2 kopecks + 3 kopecks = 5 kopecks (?). If at least one of these equalities does not hold (the scales are not in balance), then the fact being checked is false. If the equalities hold, we will perform two more weighings to check the following equalities: 1 kopeck + 3 kopecks + 5 kopecks = 9 grams (?), and 2 kopecks + 3 kopecks + 5 kopecks = 1 kopeck + 9 grams (?). If at least one of these equalities does not hold, then the fact being checked is false. If both equalities hold, we will have two equations: \( x + x + y + x + 2y = 9 \) and \( y + x + y + x + 2y = 9 + x \), from which we get \( x = 1, y = 2 \), i.e., the fact being checked is true.