2002 AMC 10A Problems/Problem 22
Problem
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: 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 1
The pattern is quite simple to see after listing a couple of terms.
Thus, the answer is .
Solution 2
Given tiles, a step removes tiles, leaving tiles behind. Now, , so in the next step tiles are removed. This gives , another perfect square.
Thus each two steps we cycle down a perfect square, and in steps, we are left with tile, hence our answer is .
Solution 3
We start of with numbers. When we use the certain operation, call if , have . Then we do again, to subtract numbers and get . In the end, we will want . We can say we have to use once to make into . Thus we must use it twice to get from to . For example, it takes us of to get from to . Then of to get from to . You should try this with and , and see it works. This means we can have be the number we start with, and be the number we want. Then we would use , times to get . Substituting for we get . - Wiselion
Video Solution
https://www.youtube.com/watch?v=CuKko0JpIdQ ~David
See also
2002 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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