Problems

Given a board (divided into squares) of the size: a) $10\times 12$, b) $9\times 10$, c) $9\times 11$, consider the game with two players where: in one turn a player is allowed to cross out any row or any column if there is at least one square not crossed out. The loser is the one who cannot make a move. Is there a winning strategy for one of the players?

A journalist came to a company which had $N$ people. He knows that this company has a person $Z$, who knows all the other members of the company, but nobody knows him. A journalist can address each member of the company with the question: “Do you know such and such?” Find the smallest number of questions sufficient to surely find $Z$. (Everyone answers the questions truthfully. One person can be asked more than one question.)

Each of the three cutlets should be fried in a pan on both sides for five minutes each side. Only two cutlets can fit onto the frying pan. Is it possible to fry all three cutlets more quickly than in 20 minutes (if the time to turn over and transfer the cutlets is neglected)?

Your task is to find out a five-digit phone number, asking questions that can be answered with either “yes” or “no.” What is the smallest number of questions for which this can be guaranteed (provided that the questions are answered correctly)?

What weights can three weights have so that they can weigh any integer number of kilograms from 1 to 10 on weighing scales (weights can be put on both cups)? Give an example.

The rook stands on the square a1 of a chessboard. For a move, you can move it by any number of cells to the right or up. The one who puts the rook on the h8 square will win. Who wins with the right strategy?

Hannah has 10 employees. Each month, Hannah raises the salary by 1 pound of exactly nine of her employees (of her choice). How can Hannah raise the salaries to make them equal? (Salaries are an integer number of pounds.)

Three friends decide, by a coin toss, who goes to get the juice. They have one coin. How do they arrange coin tosses so that all of them have equal chances to not have to go and get the juice?

There are 8 glasses of water on the table. You are allowed to take any two of the glasses and make them have equal volumes of water (by pouring some water from one glass into the other). Prove that, by using such operations, you can eventually get all the glasses to contain equal volumes of water.

A broken calculator carries out only one operation “asterisk”: $a\ast b=1-a/b$. Prove that using this calculator it is possible to carry out all four arithmetic operations (addition, subtraction, multiplication, division).