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Perimeter is an Integer

Source: 2015 AMC 12A #19/10A #24

February 4, 2015
geometryperimetercalculusrhombusconicsparabola

Problem Statement

For some positive integers pp, there is a quadrilateral ABCDABCD with positive integer side lengths, perimeter pp, right angles at BB and CC, AB=2AB=2, and CD=ADCD=AD. How many different values of p<2015p<2015 are possible?
<spanclass=′latex−bold′>(A)</span>30<spanclass=′latex−bold′>(B)</span>31<spanclass=′latex−bold′>(C)</span>61<spanclass=′latex−bold′>(D)</span>62<spanclass=′latex−bold′>(E)</span>63<span class='latex-bold'>(A) </span>30\qquad<span class='latex-bold'>(B) </span>31\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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