Problem:
A rectangular piece of paper whose length is times the width has area . The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area . What is the ratio ?
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Solution:
Without loss of generality, assume that the rectangle has dimensions by . Then the fold has length , and the overlapping areas are equilateral triangles each with area . The new shape has area , and the desired ratio is .
The problems on this page are the property of the MAA's American Mathematics Competitions