1993 AIME Problems/Problem 9
Two thousand points are given on a circle. Label one of the points . From this point, count points in the clockwise direction and label this point . From the point labeled , count points in the clockwise direction and label this point . (See figure.) Continue this process until the labels are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as ?
The label will occur on the th point around the circle. (Starting from 1) A number will only occupy the same point on the circle if .
Simplifying this expression, we see that . Therefore, one of or is odd, and each of them must be a multiple of or .
For to be a multiple of and to be a multiple of , and . The smallest for this case is .
In order for to be a multiple of and to be a multiple of , and . The smallest for this case is larger than , so is our answer.
Note: One can just substitute and to simplify calculations.
Two labels and occur on the same point if . If we assume the final answer be , then we have .
Multiply on both side we have . As they have different parities, the even one must be divisible by . As , one of them is divisible by , which indicates it's divisible by .
Which leads to four different cases: ; ; and ; and . Which leads to and respectively, and only satisfied.Therefore answer is .(by ZJY)
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