6. Given a rectangle of size $2021 \times 4300$. Inside it, there is a billiard ball. It is launched in a straight line, forming a $45^{\circ}$ angle with the sides of the rectangle. Upon reaching a side, the ball reflects at a $45^{\circ}$ angle; if the ball hits a corner, it exits along the same line it entered. (An example of the beginning of the ball's path is shown in the figure.)

a) Is it true for any point that if a ball is launched from it according to these rules, it will return to that point again?

b) Suppose that starting from some point $A$, the ball returns to it after some time. What is the maximum number of bounces off the sides it can make before returning to point $A$ for the first time?

(O. A. Plyve)

![](https://cdn.mathpix.com/cropped/2024_05_06_7ac8905f0a14a1578722g-2.jpg?height=487&width=255&top_left_y=2255&top_left_x=1666)
# Solution.

We will replicate the rectangle multiple times by reflecting it relative to its sides. In adjacent columns (rows), copies of the rectangle will be oriented differently, but when shifted by an even number of columns and an even number of rows, the orientation will match the initial one. Now, we can consider that when the ball touches the edge, it transitions to the next copy of the rectangle but continues along the same straight line. We are interested in the moment when the ball reaches one of the copies of the initial point.

a) Let's introduce a coordinate system such that one of the corners of the board has coordinates \((0,0)\), the horizontal side is shorter than the vertical side, and at the beginning of the ball's movement

![](https://cdn.mathpix.com/cropped/2024_05_06_7ac8905f0a14a1578722g-3.jpg?height=708&width=600&top_left_y=109&top_left_x=1299)
both coordinates increase.

If the coordinates of the initial point are \(\left(x_{0}, y_{0}\right)\), then any point on the path has coordinates \(\left(x_{0}+a, y_{0}+a\right)\). A shift by \(a=\operatorname{LCM}(2021 \cdot 2, 4300 \cdot 2)\) along each coordinate brings us to a copy of the initial point, since we have moved an even number of sides of the rectangle in each direction.

b) Note that \(2021 = 43 \cdot 47\) and \(4300 = 43 \cdot 100\), so \(\operatorname{LCM}(2021 \cdot 2, 4300 \cdot 2) = 2 \cdot 43 \cdot 47 \cdot 100\). A shift by \(a = 2 \cdot 43 \cdot 47 \cdot 100\) along each coordinate corresponds to a shift of \(2 \cdot 100\) sides of length 2021 to the right and \(2 \cdot 47\) sides of length 4300 upwards. This means that a total shift of 294 rectangles to the right and upwards will bring us to a copy of the initial point. Each transition to the next rectangle corresponds to one bounce of the ball, so the coincidence will occur after 294 bounces.

Note that a coincidence with a copy of the initial point in a rectangle oriented as the initial one cannot occur earlier, since \(a\) must divide both \(2 \cdot 2021\) and \(2 \cdot 4300\). However, the number of bounces before the first coincidence can be less for one of the following two reasons:

- the trajectory passes through a corner (then two bounces "merge" into one);
- the ball hits the initial point in one of the copies oriented differently than the initial rectangle.

Let's provide an example where this does not happen: let \(x_{0} = \sqrt{3}\) and \(y_{0} = \sqrt{2}\). The trajectory will not pass through a corner, since for this \(x - y\) must be an integer, while it is always \(\sqrt{3} - \sqrt{2}\). Let's study the question of copies of the point in "reflected" rectangles. The coordinates of copies of the initial points always take the form \(\left(2021 k \pm x_{0}, 4300 l \pm y_{0}\right)\), where the choice of signs depends on the parity of \(k\) and \(l\), and in "reflected" rectangles at least one of the signs is minus. Suppose, for example, \(x_{0} + a = 2021 k - x_{0}\), then \((a + 2 \sqrt{2})\) is an integer. Depending on the sign in the second coordinate, we get that \(a\) or \((a + 2 \sqrt{3})\) is an integer, but this is impossible.

Answer: a) yes b) 294 bounces.

Criteria. Part a - 2 points, part b - 5 points. Correct answer to part b - 1 point.