Problems

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Alice the fox and Basilio the cat have grown \(20\) counterfeit bills on a money tree and now write seven-digit numbers on them. Each bill has \(7\) empty cells for numbers. Basilio calls out one digit "1" or "2" (he doesn’t know the others), and Alice writes the number into any empty cell of any bill and shows the result to Basilio. When all the cells are filled, Basilio takes as many bills with different numbers as possible (out of several with the same number, he takes only one), and the rest is taken by Alice. What is the largest number of bills Basilio can get, regardless of Alice’s actions?

Detective Nero Wolf investigates a crime. He’s got \(80\) people involved in the case, among whom one is a criminal and another is a witness to the crime (but it is not known who either of them are). Each day the detective may invite one or more of these \(80\) people, and if there is a witness among those invited, but not the perpetrator, the witness will report who the perpetrator is. Can the detective solve a case in \(12\) days?

There are \(16\) cities in the kingdom. Prove that it is not possible to build a system of roads in such a way that one can get from any city to any other without passing through more than one city on the way, and with at most four roads coming out of each city.