Iran MO Round 2 2019 P1 wrote: We have a rectangle with it sides being a mirror.A light Ray enters from one of the corners of the rectangle and after being reflected several times enters to the opposite corner it started.Prove that at some time the light Ray passed the center of rectangle(Intersection of diagonals.) Solution: Let $ABCD$ be the given rectangle. Suppose, the Ray bounces off the sidelines $n$ times at points $X_1,X_2, \ldots$ $,X_n$. If $n$ were odd, then $$AX_1||X_2X_3||X_4X_5 || \ldots || X_{n-1}X_n$$Hence, we would have, $X_1X_2||X_nC$. Now, let the reflections of $A$ over $B,D$ be $B',D'$. Suppose, $X_1 \in \overline{BC}$ $\implies$ $X_1B'$ $||$ $CX_n$ which implies $X_n$ lies outside $ABCD$, A Contradiction! Similarly, If $X_1$ $\in$ $\overline{CD}$, we again get a contradiction! Hence, $n$ must be even. WLOG, Assume $X_1$ $\in$ $\overline{CD}$. The other case can be dealt similarly. Let the $AC \cap BD=M$. Now, $AX_1CX_n$ is a Parallelogram. Let $\stackrel{M}{\mapsto}$ denote homothety at $M$ with factor of $-1$. Then, $X_1 \stackrel{M}{\mapsto} X_n$. Now, Since, $\overline{AX_1}$ $\stackrel{M}{\mapsto}$ $\overline{CX_n}$ $\implies$ $\overline{X_1X_2}$ $\stackrel{M}{\mapsto}$ $\overline{X_nX_{n-1}}$ $\implies$ $\overline{X_2X_3}$ $\stackrel{M}{\mapsto}$ $\overline{X_{n-1}X_{n-2}}$. This can be continued and can be expressed: $$\overline{X_kX_{k+1}} \stackrel{M}{\mapsto} \overline{X_{n+1-k}X_{n-k}}$$So, when $k=\tfrac{n}{2}$ $\implies$ $\overline{X_{n/2}X_{(n/2)+1}}$ $\stackrel{M}{\mapsto}$ $\overline{X_{(n/2)+1}X_{n/2}}$ $\implies$ $M$ $\in$ $\overline{X_{n/2}X_{(n/2)+1}}$ $\qquad \blacksquare$