Difference between revisions of "1997 AIME Problems"

 
(Problem 7)
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[[1997 AIME Problems/Problem 7|Solution]]
 
[[1997 AIME Problems/Problem 7|Solution]]
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There is a set of 1000 switches, each of which has four positions, called <math>A, B, C</math>, and <math>D</math>.  When the position of any switch changes, it is only from <math>A</math> to <math>B</math>, from <math>B</math> to <math>C</math>, from <math>C</math> to <math>D</math>, or from <math>D</math> to <math>A</math>.  Initially each switch is in position <math>A</math>.  The switches are labeled with the 1000 different integers <math>(2^{x})(3^{y})(5^{z})</math>, where <math>x, y</math>, and <math>z</math> take on the values <math>0, 1, \ldots, 9</math>.  At step i of a 1000-step process, the <math>i</math>-th switch is advanced one step, and so are all the other switches whose labels divide the label on the <math>i</math>-th switch.  After step 1000 has been completed, how many switches will be in position <math>A</math>?
  
 
== Problem 8 ==
 
== Problem 8 ==

Revision as of 17:42, 12 August 2006

Problem 1

Solution

Problem 2

Solution

Problem 3

Solution

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution There is a set of 1000 switches, each of which has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to $D$, or from $D$ to $A$. Initially each switch is in position $A$. The switches are labeled with the 1000 different integers $(2^{x})(3^{y})(5^{z})$, where $x, y$, and $z$ take on the values $0, 1, \ldots, 9$. At step i of a 1000-step process, the $i$-th switch is advanced one step, and so are all the other switches whose labels divide the label on the $i$-th switch. After step 1000 has been completed, how many switches will be in position $A$?

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

See also

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