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| ==Problem==
| | #redirect [[2002 AMC 12A Problems/Problem 15]] |
| 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?
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| <math>\text{(A)}\ 10 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qquad \text{(E)}\ 20</math>
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| ==Solution==
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| Given <math>n^2</math> tiles, a step removes <math>n</math> tiles, leaving <math>n^2 - n</math> tiles behind. Now, <math>(n - 1)^2 = n^2 - n + (1 - n) < n^2 - n < n^2</math>, so in the next step <math>n - 1</math> tiles are removed. This gives <math>(n^2 - n) - (n - 1) = n^2 - 2n + 1 = (n - 1)^2</math>, another perfect square, and the process repeats.
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| Thus each two steps we cycle down a perfect square, and in <math>(10 - 1)\times 2 = 18</math> steps, we are left with <math>1</math> tile.
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| ==See Also==
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| {{AMC10 box|year=2002|ab=A|num-b=20|num-a=22}}
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| [[Category:Introductory Combinatorics Problems]] | |
