ToT 2025-spring JA5

Given $15$ coins of the same appearance. It is known that one of them weighs $1$ g, two coins weigh $2$ g each, three coins weigh $3$ g each, four coins weigh $4$ g each, and five coins weigh $5$ g each. There are inscriptions on the coins, indicating their weight. It is allowed to perform two weighings on a balance without additional weights. Find a way to check that there are no wrong inscriptions. (It is not required to check which inscriptions are wrong and which ones are correct.) (8 marks)