¶ 2015 AMC 10A Problem 24
Problem:
For some positive integers , quadrilateral with positive integer side lengths has perimeter , right angles at and , and . How many different values of are possible?
Answer Choices:
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E.
Solution:
In every such quadrilateral, . Let be the foot of the perpendicular from to ; then and . Let and ; then . By the Pythagorean Theorem, , or . Therefore is even, say , and . The perimeter of the quadrilateral is . Increasing positive integer values of give the required quadrilaterals, with increasing perimeter. For the perimeter is , and for the perimeter is . Therefore there are such quadrilaterals.
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The problems on this page are the property of the MAA's American Mathematics Competitions