Problem 993

Solution:

The ball travels a horizontal distance of $1000$ units between two bounces from the sides $AB$ and $CD$ as it always moves on a line making a $45^{\circ}$ angle with the sides. Hence it is always at a distance of even number of units to the line $AD$ when it hits $AB$ or $CD$. Hence it can never hit $AB$ at $B$.

Solution:

Consider a rectangle $A'B'C'D'$ which is wider $1/2$ units on all sides, divide it into unit squares, and color them black and white alternatingly with the vertex $A$ being the center of a black unit square. Then the ball always moves along the diagonals of the black unit squares. As $B$ lies at the center of a white unit square, the ball never reaches $B$.

Solution:

The vertical lines $x = 2013m$ and the horizontal lines $y = 1000n$, where $m$ and $n$ are integers, divide the $xy$-plane into rectangles congruent to the rectangle $ABCD$. Let $A(0,0)$, $B(2013,0)$, $C(2013,1000)$, $D(0,1000)$, and identify the other rectangles with $ABCD$ via reflections across these lines. Under this identification, the ball moves along the line $y = x$ and the coordinates of the points identified with $B$ have the form $(2013k, 1000l)$ where $k$ is an odd integer and $l$ is an even one. Hence the ball never goes to $B$.