2025 AMC 10A Problem 20

Problem 20 of 25HarderAlgebra Geometry

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?

Show solution

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

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.