Difference between revisions of "1999 AIME Problems/Problem 7"
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Revision as of 18:54, 19 September 2007
Problem
There is a set of 1000 switches, each of which has four positions, called , and
. When the position of any switch changes, it is only from
to
, from
to
, from
to
, or from
to
. Initially each switch is in position
. The switches are labeled with the 1000 different integers
, where
, and
take on the values
. At step i of a 1000-step process, the
-th switch is advanced one step, and so are all the other switches whose labels divide the label on the
-th switch. After step 1000 has been completed, how many switches will be in position
?
Solution
For each th switch (designated by
), it advances itself only one time at the
th step; thereafter, only a switch with larger
values will advance the
th switch by one step provided
divides
. Let
be the max switch label. To find the divisor multiples in the range of
to
, we consider the exponents of the number
. In general, the divisor-count of
must be a multiple of 4 to ensure that a switch is in position A:
, where
We consider the cases where the 3 factors above do not contribute multiples of 4.
Case of no 2's:
There are odd integers in
to
.
We have ways.
Case of a single 2:
or
or
Since the terms
and
are three valid choices for the
factor above.
We have ways.
The number of switches in position A is:
.
See also
1999 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |