Difference between revisions of "2002 AMC 10A Problems/Problem 21"
(this solution is due to azjps) |
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Revision as of 23:41, 26 December 2008
Problem
A set of tiles numbered 1 through 100 is modified repeatedly as follows: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?
Solution
Given tiles, a step removes
tiles, leaving
tiles behind. Now,
, so in the next step
tiles are removed. This gives
, another perfect square, and the process repeats.
Thus each two steps we cycle down a perfect square, and in steps, we are left with
tile.
See Also
2002 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |