[ Symmetric Strategy ]

On the board, two numbers are written: 2014 and 2015. Petya and Vasya take turns, with Petya starting first. In one move, they can

- either decrease one of the numbers by its non-zero digit or by a non-zero digit of the other number;
- or divide one of the numbers by two if it is even.

The player who first writes a single-digit number wins. Who among them can win, regardless of how the opponent plays?
Let Petya first replace 2015 with 2014, and with each subsequent move, he will equalize the numbers (he can always do this by repeating Vasya's move with the number Vasya did not change):

![](https://cdn.mathpix.com/cropped/2024_05_06_568737668fd6fdc3aecdg-02.jpg?height=203&width=889&top_left_y=1&top_left_x=590)

If Petya acts this way throughout the game, then, of course, at some point Vasya will make one of the two identical numbers a single-digit number and win.

But let's look at this moment more closely. If Vasya wins by replacing one of the two numbers \(X\) in the pair \((X, X)\) with a single-digit number, then before this, on Petya's turn, the number \(X\) was already on the board. At this moment, Petya can replace \(X\) with a single-digit number and win:

![](https://cdn.mathpix.com/cropped/2024_05_06_568737668fd6fdc3aecdg-02.jpg?height=328&width=778&top_left_y=459&top_left_x=663)

(Petya can do this because he has all the options that Vasya had on the last, winning move: dividing the number \(X\) in half if it is even, and subtracting its digit from it.)

Thus, let's formulate Petya's strategy completely: "if one of the numbers can be replaced with a single-digit number - do it; otherwise, equalize the two numbers."

## Answer

Petya.

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Author: Buhretov A.i.

Misha stands at the center of a circular lawn with a radius of 100 meters. Every minute, he takes a step of 1 meter. Before each step, he announces the direction in which he wants to step. Katya has the right to force him to change direction to the opposite. Can Misha act in such a way that he will definitely step off the lawn at some point, or can Katya always prevent him?

## Solution

After the \(n\)-th move, Misha can be exactly \(\sqrt{n}\) meters away from the center. For this, he only needs to choose a direction perpendicular to the direction of the segment connecting the point where he stands to the center each time.

## Answer

He can.



保留了原文的换行和格式。