Question

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 $g \gt 0$ 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 $\dfrac{a\sqrt{b} - c}{d},$ 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 $a + b + c + d?$

$119$

$120$

$121$

$122$

$123$

Answer: A

Concepts:tangent line coordinate geometry quadratic

Difficulty rating: 2080

Solution:

Put the silo's center at the origin with radius $10.$ Then MacDonald is at $D = (-20, -15)$ and McGregor at $G = (20, -g).$ The tangent length from $D$ is $DT = \sqrt{DS^2 - 10^2}$ $= \sqrt{25^2 - 100}$ $= \sqrt{525},$ and from $G$ it is $TG = \sqrt{g^2 + 20^2 - 10^2}.$ The tangent point $T$ sits between the two men, so $DG = DT + TG.$ But also $DG = \sqrt{40^2 + (15 - g)^2}.$ Squaring twice and simplifying gives $3g^2 + 150g - 925 = 0.$ Its positive root, which satisfies the original tangent-length equation, is $g = \frac{20\sqrt{21} - 75}{3}.$ Therefore $a + b + c + d$ $= 20 + 21 + 75 + 3$ $= 119.$ Thus, A is the correct answer.

2025 AMC 10A Problem 20

Answer: A

Difficulty rating: 2080

Solution:

Problem 20 in Other Years