Problem:
A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is foot wide on all four sides. What is the length in feet of the inner rectangle?
Answer Choices:
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Solution:
Let the inner rectangle's length be feet; then its area is square feet. The middle region has area , so the difference in the arithmetic sequence is equal to . The outer region has area , so the difference in the arithmetic sequence is also equal to . From , it follows that . The regions have areas , and .
The problems on this page are the property of the MAA's American Mathematics Competitions