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2008/09 AHSMC Part 1 Problem 6

Problem 6 of 16

A building is in the shape of a right circular cylinder, with mirrors for its interior wall. A horizontal ray starts from a point on the wall, reflects off the wall exactly 1111 times and returns to its starting point for the first time. The number of possible directions along which it may have started is

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Solution

Consider a horizontal cross-section of the room as a clock, and let the ray start at the 1212 o'clock position. It must travel directly at another point on the clock in order to return to its starting position after 1111 bounces. However, some of these 1111 paths take it back to the starting point before 1111 bounces. The initial destination must be a point which is relatively prime to 1212, namely, 11, 55, 77 or 1111. The answer is (A).

University of Alberta