Mock AIME 4 2006-2007 Problems/Problem 2

Problem

Two points $A(x_1, y_1)$ and $B(x_2, y_2)$ are chosen on the graph of $f(x) = \ln x$ , with $0 < x_1 < x_2$ . The points $C$ and $D$ trisect $\overline{AB}$ , with $AC < CB$ . Through $C$ a horizontal line is drawn to cut the curve at $E(x_3, y_3)$ . Find $x_3$ if $x_1 = 1$ and $x_2 = 1000$ .

Since $C$ is the trisector of line segment $\overline{AB}$ closer to $A$ , the $y$ -coordinate of $C$ is equal to two thirds the $y$ -coordinate of $A$ plus one third the $y$ -coordinate of $B$ . Thus, point $C$ has coordinates $(x_0, \frac{2}{3} \ln 1 + \frac{1}{3}\ln 1000) = (x_0, \ln 10)$ for some $x_0$ . Then the horizontal line through $C$ has equation $y = \ln 10$ , and this intersects the curve $y = \ln x$ at the point $(10, \ln 10)$ , so $x_3 = 10$ .