Difference between revisions of "2006 AIME A Problems/Problem 11"
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== Problem == | == Problem == | ||
− | A [[ | + | A collection of 8 [[cube (geometry) | cube]]s consists of one cube with [[edge]]-[[length]] <math> k </math> for each [[integer]] <math> k, 1 \le k \le 8. </math> A tower is to be built using all 8 cubes according to the rules: |
+ | |||
+ | * Any cube may be the bottom cube in the tower. | ||
+ | * The cube immediately on top of a cube with edge-length <math> k </math> must have edge-length at most <math> k+2. </math> | ||
+ | |||
+ | Let <math> T </math> be the number of different towers than can be constructed. What is the [[remainder]] when <math> T </math> is divided by 1000? | ||
== Solution == | == Solution == |
Revision as of 15:43, 25 September 2007
Problem
A collection of 8 cubes consists of one cube with edge-length for each integer A tower is to be built using all 8 cubes according to the rules:
- Any cube may be the bottom cube in the tower.
- The cube immediately on top of a cube with edge-length must have edge-length at most
Let be the number of different towers than can be constructed. What is the remainder when is divided by 1000?
Solution
Define the sum as . Notice that , so the sum will be:
The first two groupings almost completely cancel. The third resembles .
and are both given; the last four digits of the sum is , and half of that is . Therefore, the answer is .
See also
2006 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |