Three people play table tennis, and the player who lost the game gives way to the player who did not participate in it. As a result, it turned out that the first player played 10 games and the second played 21 games. How many games did the third player play?
Prove that multiplying the polynomial \((x + 1)^{n-1}\) by any polynomial different from zero, we obtain a polynomial having at least \(n\) nonzero coefficients.
Solve the equation \(xy = x + y\) in integers.
Determine all solutions of the equation \((n + 2)! - (n + 1)! - n! = n^2 + n^4\) in natural numbers.
10 numbers are written around the circle, the sum of which is equal to 100. It is known that the sum of every three numbers standing side by side is not less than 29.
Specify the smallest number \(A\) such that in any such set of numbers each of the numbers does not exceed \(A\).
A daisy has a) 12 petals; b) 11 petals. Consider the game with two players where: in one turn a player is allowed to remove either exactly one petal or two petals which are next to each other. The loser is the one who cannot make a turn. How should the second player act, in cases a) and b), in order to win the game regardless of the moves of the first player?
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?
In a group of friends, each two people have exactly five common acquaintances. Prove that the number of pairs of friends is divisible by 3.
At a round table, 10 boys and 15 girls were seated. It turned out that there are exactly 5 pairs of boys sitting next to each other.
How many pairs of girls are sitting next to each other?
Of 11 balls, 2 are radioactive. For any set of balls in one check, you can find out if there is at least one radioactive ball in it (but you cannot tell how many of them are radioactive). Is it possible to find both radioactive balls in 7 checks?