Problems

A monkey escaped from it’s cage in the zoo. Two guards are trying to catch it. The monkey and the guards run along the zoo lanes. There are six straight lanes in the zoo: three long ones form an equilateral triangle and three short ones connect the middles of the triangle sides. Every moment of the time the monkey and the guards can see each other. Will the guards be able to catch the monkey, if it runs three times faster than the guards? (In the beginning of the chase the guards are in one of the triangle vertices and the monkey is in another one.)

Prove that in a game of noughts and crosses on a $3\times 3$ grid, if the first player uses the right strategy then the second player cannot win.

A rectangular chocolate bar size $5\times 10$ is divided by vertical and horizontal division lines into 50 square pieces. Two players are playing the following game. The one who starts breaks the chocolate bar along some division line into two rectangular pieces and puts the resulting pieces on the table. Then players take turns doing the same operation: each time the player whose turn it is at the moment breaks one of the parts into two parts. The one who is the first to break off a square slice $1\times 1$ (without division lines) a) loses; b) wins. Which of the players can secure a win: the one who starts or the other one?

There is a system of equations $$\begin{array}{rl}\ast x+\ast y+\ast z& =0,\\ \ast x+\ast y+\ast z& =0,\\ \ast x+\ast y+\ast z& =0.\end{array}$$ Two people alternately enter a number instead of a star. Prove that the player that goes first can always ensure that the system has a non-zero solution.

Two players play on a square field of size $99\times 99$, which has been split onto cells of size $1\times 1$. The first player places a cross on the center of the field; After this, the second player can place a zero on any of the eight cells surrounding the cross of the first player. After that, the first puts a cross onto any cell of the field next to one of those already occupied, etc. The first player wins if he can put a cross on any corner cell. Prove that with any strategy of the second player the first can always win.

Two players in turn paint the sides of an $n$-gon. The first one can paint the side that borders either zero or two colored sides, the second – the side that borders one painted side. The player who can not make a move loses. At what $n$ can the second player win, no matter how the first player plays?

The Hatter plays a computer game. There is a number on the screen, which every minute increases by 102. The initial number is 123. The Hatter can change the order of the digits of the number on the screen at any moment. His aim is to keep the number of the digits on the screen below four. Can he do it?

Show that in the game “Noughts and Crosses” the second player never wins if the first player is smart enough.

There is a chequered board of dimension $10\times 12$. In one go you are allowed to cross out any row or column if it contains at least one square which was not crossed out yet. The loser is the player who cannot make a move. Is there a winning strategy for any player?

Pathways in the Wonderland zoo make a equilateral triangles with middle lines drawn. A monkey has escaped from it’s cage. Two zoo caretakers are catching the monkey. Can zookeepers catch the monkey if all three of them are running only on pathways, the running speeds of the monkey and the zookeepers are equal, and they are all able to see each other?