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Triangle Inequality with Angle Bisectors

Source: 2018 AMC 12B #12

February 16, 2018
geometryangle bisectorAMCAMC 122018 AMC 12B2018 AMC

Problem Statement

Side AB‾\overline{AB} of △ABC\triangle ABC has length 1010. The bisector of angle AA meets BC‾\overline{BC} at DD, and CD=3CD = 3. The set of all possible values of ACAC is an open interval (m,n)(m,n). What is m+nm+n?
<spanclass=′latex−bold′>(A)</span>16<spanclass=′latex−bold′>(B)</span>17<spanclass=′latex−bold′>(C)</span>18<spanclass=′latex−bold′>(D)</span>19<spanclass=′latex−bold′>(E)</span>20 <span class='latex-bold'>(A) </span>16 \qquad <span class='latex-bold'>(B) </span>17 \qquad <span class='latex-bold'>(C) </span>18 \qquad <span class='latex-bold'>(D) </span>19 \qquad <span class='latex-bold'>(E) </span>20 \qquad
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