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Option 1

1)      Two players play the following game. On the coordinate plane at the point with coordinates (1; 2) there is a chip. Players take turns. The move consists in the fact that the player moves the piece from a point with coordinates (x, y) to one of three points: (x + 3; y), (x, y + 3) or (x, y + 4). The winner is the player, after whose move the distance in a straight line from the chip to the origin (0,0) is more than 13 units. Who wins - the player who moves first or the player who moves second?

2)      Two players play the following game. In front of them are two piles of stones, in the first of which there are 3, and in the second - 6 stones. Each player has an unlimited number of stones. Players take turns. The move consists in the fact that the player either doubles the number of stones in some heap, or adds 2 stones to some heap. The player wins, after whose move the total number of stones in the two piles becomes at least 24. Who wins if both players play correctly - the player who makes the first move or the player who makes the second move? What should be the winning player's first move? Justify the answer.

3)      Additionally

Two players play the following game. In front of them are two piles of stones, in the first of which there are 4 stones, and in the second - 5 stones. Each player has an unlimited number of stones. Players take turns. The move consists in the fact that the player either doubles the number of stones in a pile, or increases the number of stones in one of the piles by 2. The player, after whose move the total number of stones in two piles becomes more than 22, loses. When both players play flawlessly, who wins - the player making the first move or the player making the second move? What should be the winning player's first move? Justify the answer.

Option 2

1)      Two players play the following game. On the coordinate plane at the point with coordinates (1; 0) there is a token. Players take turns. The move consists in the fact that the player moves the piece from a point with coordinates (x, y) to one of three points: (x + 4; y), (x, y + 4) or (x + 4, y + 4). The winner is the player, after whose move the distance in a straight line from the chip to the origin (0,0) is more than 13 units. Who wins - the player who moves first or the player who moves second?

2)      Two players play the following game. In front of them are two piles of stones, in the first of which 1, and in the second - 4 stones. Each player has an unlimited number of stones. Players take turns. The move consists in the fact that the player either increases 3 times the number of stones in some pile, or adds 3 stones to some pile. The player wins after whose move the total number of stones in the two piles becomes at least 22. Who wins if both players play without errors - the player who makes the first move, or the player who makes the second move? What should be the winning player's first move? Justify the answer.

3)      Additionally

Two players play the following game. In front of them are two piles of stones, in the first of which there are 3 stones, and in the second - 5 stones. Each player has an unlimited number of stones. Players take turns. The move consists in the fact that the player either doubles the number of stones in a pile, or increases the number of stones in one of the piles by 2. The player, after whose turn the total number of stones in the two piles becomes more than 17, loses. When both players play flawlessly, who wins - the player making the first move or the player making the second move? What should be the winning player's first move? Justify the answer.

Option 3

1)      Two players play the following game. On the coordinate plane at a point with coordinates
(-3; 2)there is a chip. Players take turns. The move consists in the fact that the player moves the piece from a point with coordinates (x, y) to one of three points: (x – 1; y + 3), (x + 3, y – 1) or (x + 2, y +2). The winner is the player, after whose move the distance in a straight line from the chip to the origin (0,0) is more than 8 units. Who wins - the player who moves first or the player who moves second?

2)      Two players play the following game. In front of them are two piles of stones, in the first of which there are 2, and in the second - 3 stones. Each player has an unlimited number of stones. Players take turns. The move consists in the fact that the player doubles or triples the number of stones in a heap. The player wins, after whose turn there are at least 20 stones in one of the piles. When both players play flawlessly, who wins - the player making the first move or the player making the second move? What should be the winning player's first move? Justify the answer.

3)      Additionally

Two players play the following game. In front of them are two piles of stones, in the first of which there are 3 stones, and in the second - 4 stones. Each player has an unlimited number of stones. Players take turns. The move consists in the fact that the player either increases the number of stones in a pile by 3 times, or increases the number of stones in one of the piles by 4. The player, after whose turn the total number of stones in two piles becomes more than 26, loses. When both players play flawlessly, who wins - the player making the first move or the player making the second move? What should be the winning player's first move? Justify the answer.

Option 4

1)      Two players play the following game. On the coordinate plane at a point with coordinates
(-1; 2)there is a chip. Players take turns. The move consists in the fact that the player moves the piece from a point with coordinates (x, y) to one of three points: (x – 1; y + 4), (x + 3, y – 1) or (x + 2, y +3). The winner is the player, after whose move the distance in a straight line from the chip to the origin (0,0) is more than 11 units. Who wins - the player who moves first or the player who moves second?

2)      Two players play the following game. In front of them are two piles of stones, in the first of which there are 1, and in the second - 2 stones. Each player has an unlimited number of stones. Players take turns. The move consists in the fact that the player either increases the number of stones in some heap by 3 times, or adds 4 stones to some heap. The player wins, after whose turn there are at least 20 stones in one of the piles. When both players play flawlessly, who wins - the player making the first move or the player making the second move? What should be the winning player's first move? Justify the answer.

3)      Additionally

Two players play the following game. In front of them are two piles of stones, in the first of which there are 3 stones, and in the second - 4 stones. Each player has an unlimited number of stones. Players take turns. The move consists in the fact that the player either increases the number of stones in a pile by 3 times, or increases the number of stones in one of the piles by 4. The player, after whose move the total number of stones in two piles becomes more than 28, loses. When both players play flawlessly, who wins - the player making the first move or the player making the second move? What should be the winning player's first move? Justify the answer.

Option 5

1)      Two players play the following game. On the coordinate plane at a point with coordinates
(3; 2)there is a chip. Players take turns. The move consists in the fact that the player moves the piece from a point with coordinates (x, y) to one of three points: (x + 3; y), (x, y + 2) or (x, y + 4). The winner is the player, after whose move the distance in a straight line from the chip to the origin (0,0) is more than 12 units. Who wins - the player who moves first or the player who moves second?

2)      Two players play the following game. In front of them are two piles of stones, in the first of which there are 2, and in the second - 3 stones. Each player has an unlimited number of stones. Players take turns. The move consists in the fact that the player either doubles the number of stones in a pile, or increases the number of stones in one of the piles by 3. The player wins, after whose move the total number of stones in the two piles becomes at least 17. Who wins if both players play without errors - the player who makes the first move or the player who makes the second move? What should be the winning player's first move? Justify the answer.

3)      Additionally

Two players play the following game. In front of them are two piles of stones, in the first of which there are 4 stones, and in the second - 5 stones. Each player has an unlimited number of stones. Players take turns. The move consists in the fact that the player either doubles the number of stones in a pile, or increases the number of stones in one of the piles by 2. The player, after whose move the total number of stones in two piles becomes more than 22, loses. When both players play flawlessly, who wins - the player making the first move or the player making the second move? What should be the winning player's first move? Justify the answer.

Option 6

1)      Two players play the following game. On the coordinate plane at a point with coordinates
(2; 3)there is a chip. Players take turns. The move consists in the fact that the player moves the piece from a point with coordinates (x, y) to one of three points: (2x; y), (x, y + 3) or (x, y + 4). The winner is the player, after whose move the distance in a straight line from the chip to the origin (0,0) is more than 14 units. Who wins - the player who moves first or the player who moves second?

2)      Two players play the following game. In front of them are two piles of stones, in the first of which there are 2, in the second - 3 stones. Each player has an unlimited number of stones. Players take turns. The move consists in the fact that the player either increases the number of stones in a pile by 3 times, or adds 3 stones to any of the piles. The player wins after whose move the total number of stones in the two piles becomes at least 33. Who wins - the player who makes the first move, or the player who makes the second move?

3)      Additionally

Two players play the following game. In front of them are two piles of stones, in the first of which there are 3 stones, and in the second - 5 stones. Each player has an unlimited number of stones. Players take turns. The move consists in the fact that the player either doubles the number of stones in a pile, or increases the number of stones in one of the piles by 2. The player, after whose turn the total number of stones in the two piles becomes more than 17, loses. When both players play flawlessly, who wins - the player making the first move or the player making the second move? What should be the winning player's first move? Justify the answer.