2006 AMC 12B Problems/Problem 20
Problem
Let be chosen at random from the interval . What is the probability that ? Here denotes the greatest integer that is less than or equal to .
Solution
Let be an arbitrary integer. For which do we have ?
The equation can be rewritten as . The second one gives us . Combining these, we get that both hold at the same time if and only if .
Hence for each integer we get an interval of values for which . These intervals are obviously pairwise disjoint.
For any the corresponding interval is disjoint with , so it does not contribute to our answer. On the other hand, for any the entire interval is inside . Hence our answer is the sum of the lengths of the intervals for .
For a fixed the length of the interval is .
This means that our result is .
See also
2006 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
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All AMC 12 Problems and Solutions |