Problem:
A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently and at random. What is the probability that the frog's final position is no more than 1 meter from its starting position?
Answer Choices:
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E.
Solution:
Let denote the frog's starting point, and let , and denote its positions after the first, second, and third jumps, respectively. Introduce a coordinate system with at at at , and at . It may be assumed that and . For , the required condition is met for all values of . For , the required condition is met only if . For if and only if or , and the required condition is met if and only if . In\
the -plane, the rectangle has area . The triangle has area , so the requested probability is .
The problems on this page are the property of the MAA's American Mathematics Competitions