Difference between revisions of "2015 AMC 12A Problems/Problem 23"

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==Solution==
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Each segment of half of the length of a side of the square is identical, so arbitrarily choose one.
  
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The portion of the square within 0.5 units of a point on that segment is 0.5+d+sqrt(.25-d^2) where d is the distance from the corner.  The integral from 0 to 0.5 of this formula resolves to 6+π/8, so the probability of choosing a point within 0.5 of the first point is 6+π/32.  The inverse of this is 26-π/32, so a+b+c=(A) 59.

Revision as of 23:48, 4 February 2015

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 0.5 units of a point on that segment is 0.5+d+sqrt(.25-d^2) where d is the distance from the corner. The integral from 0 to 0.5 of this formula resolves to 6+π/8, so the probability of choosing a point within 0.5 of the first point is 6+π/32. The inverse of this is 26-π/32, so a+b+c=(A) 59.

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