Circle 12 05 (Lesson1) – We Solve Problems

#### Circle 12 05 (Lesson1)

Circle name: Circle 12 05
Lesson name: Lesson1
Starts at : 13.05.2020 09:00

Problems:

#### Game theory (other) , Theory of algorithms (other)10-12

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?
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A                                                       B                                                          C

#### Sequences15-18

For which natural $n$ does the number $\frac{n^2}{1,001^n}$ reach its maximum value?

#### Functional equations15-18

The function F is given on the whole real axis, and for each $x$ the equality holds: $F (x + 1) F (x) + F (x + 1) + 1 = 0.$
Prove that the function F can not be continuous.

#### Chessboards and chess pieces , Game theory10-12

Two play a game on a chessboard $8 × 8.$ 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?

#### Game theory (other) , Theory of algotithms10-13

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?

#### Regular polygons , Symmetric strategies12-14

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.

#### Integer and fractional parts. Archimedean property14-16

a$)$ Give an example of a positive number a such that ${a} + {1 / a} = 1.$
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b$)$ Can such an a be a rational number?

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