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Three Real Numbers Chosen From [0,n]

Source: 2012 AMC10A Problem #25

February 8, 2012
probabilitygeometrysymmetryintegrationcalculus

Problem Statement

Real numbers x,yx,y, and zz are chosen independently and at random from the interval [0,n][0,n] for some positive integer nn. The probability that no two of x,yx,y, and zz are within 11 unit of each other is greater than 12\tfrac{1}{2}. What is the smallest possible value of nn?
<spanclass=′latex−bold′>(A)</span> 7<spanclass=′latex−bold′>(B)</span> 8<spanclass=′latex−bold′>(C)</span> 9<spanclass=′latex−bold′>(D)</span> 10<spanclass=′latex−bold′>(E)</span> 11 <span class='latex-bold'>(A)</span>\ 7 \qquad<span class='latex-bold'>(B)</span>\ 8 \qquad<span class='latex-bold'>(C)</span>\ 9 \qquad<span class='latex-bold'>(D)</span>\ 10 \qquad<span class='latex-bold'>(E)</span>\ 11
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