2015 AIME II Problems/Problem 4

Problem

In an isosceles trapezoid, the parallel bases have lengths $\log 3$ and $\log 192$ , and the altitude to these bases has length $\log 16$ . The perimeter of the trapezoid can be written in the form $\log 2^p 3^q$ , where $p$ and $q$ are positive integers. Find $p + q$ .

Solution

Call the trapezoid $ABCD$ with $AB$ as the smaller base and $CD$ as the longer. The point where an altitude intersects the larger base be $E$ where $E$ is closer to $D$ .

Subtract the two bases and divide to find that $ED$ is $\log 8$ . The altitude can be expressed as $\frac{4}{3} log 8$ . Therefore, the two legs are $\frac{5}{3} \log 8$ , or $\log 32$ .

The perimeter is thus $\log 32 + \log 32 + \log 192 + \log 3$ which is $\log 2^{16} 3^2$ . So $p + q = \boxed{18}$

2015 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2015 AIME II Problems/Problem 4

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