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2025 AMC 10A Problems/Problem 20: Difference between revisions

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Let the silo center be <imath>O</imath>, let the point MacDnoald is situated at be <imath>A</imath>, and let the point <imath>20</imath> meters west of the silo center be <imath>B</imath>. <imath>ABO</imath> is then a right triangle with side lengths <imath> 15, 20,</imath> and <imath>25</imath>.  
Let the silo center be <imath>O</imath>, let the point MacDnoald is situated at be <imath>A</imath>, and let the point <imath>20</imath> meters west of the silo center be <imath>B</imath>. <imath>ABO</imath> is then a right triangle with side lengths <imath> 15, 20,</imath> and <imath>25</imath>.  


Let the point <imath>20</imath> meters east of the silo center be <imath>C</imath>, and let the point McGregor is at be <imath>D</imath> with <imath>CD=g>o</imath>
Let the point <imath>20</imath> meters east of the silo center be <imath>C</imath>, and let the point McGregor is at be <imath>D</imath> with <imath>CD=g>0</imath>. <imath>AD</imath> is tangent to circle <imath>O</imath> at <imath>T</imath>,

Revision as of 15:10, 6 November 2025

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 light 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 $a+b+c+d$?

$\textbf{(A) } 118 \qquad\textbf{(B) } 119 \qquad\textbf{(C) } 120 \qquad\textbf{(D) } 121 \qquad\textbf{(E) } 122$

Solution 1

Let the silo center be $O$, let the point MacDnoald is situated at be $A$, and let the point $20$ meters west of the silo center be $B$. $ABO$ is then a right triangle with side lengths $15, 20,$ and $25$.

Let the point $20$ meters east of the silo center be $C$, and let the point McGregor is at be $D$ with $CD=g>0$. $AD$ is tangent to circle $O$ at $T$,

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