Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.2 AIME, harder Find the answer

4. There are four rectangles with length $a=18$ . Their widths form a geometric sequence. The perimeter of the second rectangle is 60, and the third rectangle is a square. Determine the widths of the rectangles.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

4. Members of a geometric sequence $b, b q, b q^{2}, b q^{3}$ ..... 1.5 points
Expression $2 a+2 b q=60$ ..... 0.5 points
$a+b q=30 \Rightarrow b q=12$ ..... 0.5 points
$a=b q^{2} \Rightarrow b q^{2}=18$ ..... 0.5 points
Solving the system ..... 1 point
$12 \cdot q=18 \Rightarrow q=\frac{3}{2}$ ..... 0.5 points
$b=8$ ..... 0.5 points
Widths of the rectangles: $8,12,18,27$ ..... 1 point

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