MathDB
Distance is Greater than One Half

Source: 2015 AMC 12A #23 / 10A #25

February 4, 2015
probabilityintegrationgeometrytrigonometrycalculusrectanglegeometric probability

Problem Statement

Let SS be a square of side length 11. Two points are chosen independently at random on the sides of SS. The probability that the straight-line distance between the points is at least 12\tfrac12 is a−bπc\tfrac{a-b\pi}c, where aa, bb, and cc are positive integers and gcd⁡(a,b,c)=1\gcd(a,b,c)=1. What is a+b+ca+b+c?
<spanclass=′latex−bold′>(A)</span>59<spanclass=′latex−bold′>(B)</span>60<spanclass=′latex−bold′>(C)</span>61<spanclass=′latex−bold′>(D)</span>62<spanclass=′latex−bold′>(E)</span>63<span class='latex-bold'>(A) </span>59\qquad<span class='latex-bold'>(B) </span>60\qquad<span class='latex-bold'>(C) </span>61\qquad<span class='latex-bold'>(D) </span>62\qquad<span class='latex-bold'>(E) </span>63
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