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Understand the problem

We have a rectangle whose sides are mirrors. A ray of light enters from one of vertices of the rectangle and after being reflected several times, exits via the vertex opposite to the initial one. Prove that at some point of time, the ray passed through the centre of rectangle (Intersection of the diagonals.)

Source of the problem
Iranian Mathematical Olympiad 2019 (second round)
Topic
Geometry/Combinatorics

Difficulty Level
Hard
It is interesting to note how Olympiads reflect the current state of mathematical research. This is not unexpected, because olympiad problems usually originate from elementary corollaries of advanced Mathematics. This particular problem has to do with the problem of billiards in the field of Dynamical Systems. The interested reader can learn more here.

Do you really need a hint? Try it first!

Instead of reflecting the ray and keeping the rectangle fixed, reflect the rectangle and keep the ray fixed.

Following hint 1, you will get a grid of rectangles and a straight line representing the path of light. If this straight line passes through one of the reflections of the opposite vertex, then in our original representation it has to pass through the opposite vertex. Similarly, if it passes through one of the reflections of the centre then in the original representation, it has to pass through the centre.

Show that the reflections of $(a,b)$ are of the form $((2m-1)a,(2n-1)b)$. Also, the reflections of the centre $(a/2,b/2)$ are of the form $((p+1/2)a,(q+1/2)b$.
Suppose that the line indeed passes through a reflection of the opposite vertex. Then it is of the form $x(t)=(2m+1)at, y(t)=(2n+1)bt$. Taking $t=\frac{1}{2}$, we see that it passes through $((m+1/2)a,(n+1/2)b)$, which is a reflection of the centre.

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