The line $y = \frac{1}{3}x + 1$ divides the square region defined by $0 \leq x \leq 2$ and $0 \leq y \leq 2$ into an upper and lower region. The line $x=a$ divides the lower region into two regions of equal area. Then $a$ can be written as $\sqrt{s} - t$, where $s$ and $t$ are positive integers. What is $s+t$?
$$\textbf{(A) } 18 \qquad \textbf{(B) } 19 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 22$$
Difficulty: 37/100 - Very algebra heavy, requires some computation.
Core Concepts: coordinates, trapezoids, analytic geometry
Challenges: Competitors who are inexperienced in finding areas in the coordinate plane will struggle. Equation solving methods may cause issues.
Techniques: Assigning coordinates to points, finding algebraic expressions for both lower areas, and creating a quadratic by equating found expressions.
Error-prone Steps: Computation errors in solving the quadratic or assuming lower regions are similar.
Ideal Time:
Experienced: ≤ 60-90 sec
Intermediate: ≤ 2-3 min
Beginner: ≤ 5 min