2015 AMC 12A Problems/Problem 23
Problem
Let be a square of side length 1. Two points are chosen independently at random on the sides of . The probability that the straight-line distance between the points is at least is , where and are positive integers and . What is ?
$\textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63$ (Error compiling LaTeX. ! Extra }, or forgotten $.)
Solution
Each segment of half of the length of a side of the square is identical, so arbitrarily choose one.
The portion of the square within units of a point on that segment is where is the distance from the corner. The integral from to of this formula resolves to , so the probability of choosing a point within of the first point is . The inverse of this is , so .
See Also
2015 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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All AMC 12 Problems and Solutions |