1993 AIME Problems/Problem 9
Problem
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
?
Solution
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.
See also
1993 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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