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.)
Problem
Source: Iranian second round/day1/problem1
Tags: geometry, rectangle, combinatorics
02.05.2019 12:34
This is not a new problem the result was known before so I had only two problems for the exam .The idea is to reflect rectangles several times so that the way that light used becomes a straight line the result is trivial then.I will add a picture to clearify what I mean. Remark:I think the problem was from russia but can't find it now.
02.05.2019 13:20
Some problems with the same base idea: Iran TST 2007, Day 2 Iran TST 2007, Day 4 APMO 2018
10.09.2019 08:48
I can't understand this language. Please translate it to English
03.10.2019 19:53
03.10.2019 20:00
There's also a PRMO question with same configuration. P24 iirc
20.10.2020 22:45
Let $O$ be the center of the rectangle and WLOG let $A$ be the starting point and let the first ray makes an angle $\alpha$ with $AD$ Claim: The path of the light is symmetric around $O$. Proof. it's easy to see that all rhymes(the lines that bisect the angle between each consecutive rays) are parallel to one of the sides so by trivial angle chasing we see that the rays always make angle $\alpha$ with $AD$ so the last ray is parallel to the first ray and since the rectangle is symmetric around $O$ then the path is also symmetric around it. $\blacksquare$ now, since this path is symmetric around $O$ and each point (without $A$ and $D$)has degree $2$ then it must pass through $O$ so we're clear
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