Difference between revisions of "2016 AIME I Problems/Problem 13"
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Revision as of 16:02, 4 March 2016
Solution
Notice that we don't really care about what the -coordinate of the frog is. So let's let denote the expected number of times Freddy will jump at a coordinate of until he reaches the line . So therefore we want to find .
So we have . Suppose Freddy is at . He has a probability of moving horizontally, chance of moving up and chance of moving down. So we have So we get the recursion . Rearranging we see . That means the difference between consecutive terms goes down by each time. So for convenience let's let and . So that means Yes, this is a quadratic. Now, notice that since there is a boundary, we have to give special care to . We have so this turns into and this results in . So now we know Now, we also have that so that gives us so . So now we know so plugging in we get
See also
2016 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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