Problems

True or false? Prince Charming went to find Cinderella. He reached the crossroads and started to daydream. Suddenly he sees the Big Bad Wolf. And everyone knows that this Big Bad Wolf on one day answers every question truthfully, and a day later he lies, he proceeds in such a manner on alternate days. Prince Charming can ask the Big Bad Wolf exactly one question, after which it is necessary for him to choose which of the two roads to go on. What question can Prince Charming ask the Big Bad Wolf to find out for sure which of the roads leads to the Magic kingdom?

In a vase, there is a bouquet of 7 white and blue lilac branches. It is known that 1) at least one branch is white, 2) out of any two branches, at least one is blue. How many white branches and how many blue are there in the bouquet?

There are 68 coins, and it is known that any two coins differ in weight. With 100 weighings on a two-scales balance without weights, find the heaviest and lightest coin.

A cat tries to catch a mouse in labyrinths A, B, and C. The cat walks first, beginning with the node marked with the letter “K”. Then the mouse (from the node “M”) moves, then again the cat moves, etc. From any node the cat and mouse go to any adjacent node. If at some point the cat and mouse are in the same node, then the cat eats the mouse.

Can the cat catch the mouse in each of the cases A, B, C?

Two play a game on a chessboard \(8 \times 8\). The player who makes the first move puts a knight on the board. Then they take turns moving it (according to the usual rules), whilst you can not put the knight on a cell which he already visited. The loser is one who has nowhere to go. Who wins with the right strategy – the first player or his partner?

Two players in turn increase a natural number in such a way that at each increase the difference between the new and old values of the number is greater than zero, but less than the old value. The initial value of the number is 2. The winner is the one who can create the number 1987. Who wins with the correct strategy: the first player or his partner?

a) The vertices (corners) in a regular polygon with 10 sides are colored black and white in an alternating fashion (i.e. one vertex is black, the next is white, etc). Two people play the following game. Each player in turn draws a line connecting two vertices of the same color. These lines must not have common vertices (i.e. must not begin or end on the same dot as another line) with the lines already drawn. The winner of the game is the player who made the final move. Which player, the first or the second, would win if the right strategy is used?

b) The same problem, but for a regular polygon with 12 sides.

We are given 101 rectangles with integer-length sides that do not exceed 100.

Prove that amongst them there will be three rectangles \(A, B, C\), which will fit completely inside one another so that \(A \subset B \subset C\).

Initially, a natural number \(A\) is written on a board. You are allowed to add to it one of its divisors, distinct from itself and one. With the resulting number you are permitted to perform a similar operation, and so on.

Prove that from the number \(A = 4\) one can, with the help of such operations, come to any given in advance composite number.

There is a chocolate bar with five longitudinal and eight transverse grooves, along which it can be broken (in total into \(9 * 6 = 54\) squares). Two players take part, in turns. A player in his turn breaks off the chocolate bar a strip of width 1 and eats it. Another player who plays in his turn does the same with the part that is left, etc. The one who breaks a strip of width 2 into two strips of width 1 eats one of them, and the other is eaten by his partner. Prove that the first player can act in such a way that he will get at least 6 more chocolate squares than the second player.