AIME 2015 Problem 4

Geometry Level 3

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 $p + q$ .

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The answer is 18.00000.

2 solutions

Chris Callahan
Dec 5, 2015

In the picture shown the two parallel bases are segments AB and CD. The two congruent sides of the trapezoid are segments AD and BC. Segments AE and BF are both altitudes to the bases.

It is given that AB=log 3, and that CD=log 192. Since AD and BC are unknown we will say they are equal to x.

Niranjan Khanderia
Dec 3, 2015

$\text{Let the equal sides be S. X and Y the parallel sides. A the altitude.} ~~~~~\text{ O the angle at the base Y. } \\ \dfrac {Y - X} 2=S*Cos 0. ~~ A=S*SinO. ~~\implies ~~~~~~~ \color{#3D99F6}{S=\dfrac A {Sin O}},\\ And ~~ O=Tan^{ - 1} \dfrac {A} {\frac{Y - X}{2} } ~~~~~ But \dfrac {A} {\frac{Y - X}{2} } =\dfrac {log2^4} {\frac{log192 - log3}{2} }=\dfrac {log2^4} {\frac{log64}{2} }=\dfrac {log2^4} {log2^3}\\ \therefore ~ O= Tan^{ - 1}\dfrac4 3=Sin^{ - 1}\dfrac 4 5. ~~~~ \implies ~S=\dfrac{log2^4}{\frac 4 5}=log2^5.\\ \therefore ~ Perimeter = 2S+X+Y =2log2^5+log3+(log64+log3) =log\{2^{16}3^2\}= log\{2^p3^q\}\\ p + q= \Large \color{#D61F06}{18}$

AIME 2015 Problem 4

Try more problems of AIME from this set AIME 2015

The answer is 18.00000.

2 solutions

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