Question 136
A balance and $100$ coins are given. Several coins are false. The number of false coins is greater than $0$ but less than $99$. All genuine coins have equal weights, and all false coins also have equal weights. The weight of a genuine coin is greater than the weight of a false one. It is permitted to perform a weighing on the balance, paying a coin before it. Prove that one can determine a true coin for sure. (The coin to be determined should not be paid for some weighing!)