A silo (right circular cylinder) with diameter 20 meters stands in a field. MacDonald is located 20 meters west and 15 meters south of the center of the silo. McGregor is located 20 meters east and $g > 0$ meters south of the center of the silo. The line of sight between MacDonald and McGregor is tangent to the silo. The value of $g$ can be written as $\frac{a\sqrt{b}-c}{d}$, where $a,b,c,$ and $d$ are positive integers, $b$ is not divisible by the square of any prime, and $d$ is relatively prime to the greatest common divisor of $a$ and $c$. What is $a+b+c+d$?
$\textbf{(A) } 119 \qquad\textbf{(B) } 120 \qquad\textbf{(C) } 121 \qquad\textbf{(D) } 122 \qquad\textbf{(E) } 123$
Difficulty: 84/100 — Exceptionally computation-heavy, although doesn't require knowledge of advanced theorems.
Core Concepts: circles, tangents, right triangle, similarity, system of equations, quadratics
Challenges: Finding a way to construct shapes in the diagram that are easier to deal with; setting up a system of equations and solving it.
Techniques: Converting the hidden similar triangles setup to algebra; knowing how to effectively solve a complicated system of equations.
Error-prone Steps: Incorrectly simplifying the final rationalization; incorrectly solving the system; switching the legs up in the similarity condition.
Ideal Time:
Experienced: ≤ 10 min
Intermediate: ≤ 20 min
Beginner: ≤ 45 min