Question

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 $p + q.$

Answer

Solution:

Dropping altitudes from the ends of the short base, each leg is the hypotenuse of a right triangle whose legs are the altitude $\log 16 = 4\log 2$ and half the difference of the bases, $\frac{1}{2}(\log 192 - \log 3) = \frac{1}{2}\log 64 = 3 \log 2.$ By the $3$ - $4$ - $5$ ratio, each leg has length $5 \log 2.$

The perimeter is $\log 3 + \log 192 + 2 \cdot 5\log 2 = \log(3 \cdot 192) + \log 2^{10} = \log(2^6 3^2) + \log 2^{10} = \log 2^{16} 3^2,$ so $p + q = 16 + 2 = 18.$

2015 AIME II Problem 4

Solution:

Problem 4 in Other Years