14

Part of 2015 AMC 12/AHSME

Problems(2)

Sum of Logarithm Reciprocals

Source: 2015 AMC 12A #14

What is the value of

a

\frac1{\log_2a}+\frac1{\log_3a}+\frac1{\log_4a}=1

?

<span class='latex-bold'>(A) </span>9\qquad<span class='latex-bold'>(B) </span>12\qquad<span class='latex-bold'>(C) </span>18\qquad<span class='latex-bold'>(D) </span>24\qquad<span class='latex-bold'>(E) </span>36

Circle and triangle Venn diagram

Source: 2015 AMC 12B Problem 14

A circle of radius

2

A

. An equilateral triangle with side

4

has a vertex at

A

. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle?

\textbf {(A) } 8-\pi \qquad \textbf {(B) } \pi + 2 \qquad \textbf {(C) } 2\pi - \frac {\sqrt{2}}{2} \qquad \textbf {(D) } 4(\pi - \sqrt{3}) \qquad \textbf {(E) } 2\pi + \frac {\sqrt{3}}{2}