There is a table in the shape of a $8\times 5$ rectangle with four holes on its corners. After shooting a ball from points $A, B$ and $C$ on the shown paths, will the ball fall into any of the holes after 6 reflections? (The ball reflects with the same angle after contacting the table edges.) Proposed by Hirad Alipanah
Problem
Source:
Tags: IGO, Iran, geometry
brainiacmaniac31
21.09.2019 02:51
Is the ball shot at the given angles of $45^\circ$, and are there $6$ balls?
Ttugf
21.09.2019 03:18
The ball shot from B going left will fall in the top left hole after 6 reflections, I don't think any of the others do.
gamerrk1004
21.09.2019 14:19
Ttugf wrote: The ball shot from B going left will fall in the top left hole after 6 reflections, I don't think any of the others do. Well ... I think you're wrong ... The ball will be able to reach one of the holes from any of the points $A$, $B$ or $C$ ...
gamerrk1004
21.09.2019 18:54
I've made a diagram for the same ... I'm sharing it with you guys ... (I looked into the official solutions ... the answer there was "YES" !! )
Attachments:
gamerrk1004
21.09.2019 18:55
The path in purple markes the path for the ball when it starts from $C$ ... The path in green markes the path for the ball when it starts from $B$ ... The path in red markes the path for the ball when it starts from $A$ I think you'll be able to prove this with the lengths given to us in the actual figure ... If any have still has any doubts regarding this problem ... PM me
Ttugf
22.09.2019 02:02
gamerrk1004 wrote: The path in purple markes the path for the ball when it starts from $C$ ... The path in green markes the path for the ball when it starts from $B$ ... The path in red markes the path for the ball when it starts from $A$ I think you'll be able to prove this with the lengths given to us in the actual figure ... If any have still has any doubts regarding this problem ... PM me Ok, I guess I interpreted the problem as exactly 6. That solution looks right.
gamerrk1004
22.09.2019 07:02
Thanks @above ...
gamerrk1004
22.09.2019 07:03
Ttugf wrote: gamerrk1004 wrote: The path in purple markes the path for the ball when it starts from $C$ ... The path in green markes the path for the ball when it starts from $B$ ... The path in red markes the path for the ball when it starts from $A$ I think you'll be able to prove this with the lengths given to us in the actual figure ... If any have still has any doubts regarding this problem ... PM me Ok, I guess I interpreted the problem as exactly 6. That solution looks right. Actually you're right ... there was a lot of confusion in the "English" of this question out there ... I also got confused by the same