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Difference between revisions of "2024 AMC 12B Problems/Problem 20"

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==Problem 20==
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Suppose <math>A</math>, <math>B</math>, and <math>C</math> are points in the plane with <math>AB=40</math> and <math>AC=42</math>, and let <math>x</math> be the length of the line segment from <math>A</math> to the midpoint of <math>\overline{BC}</math>. Define a function <math>f</math> by letting <math>f(x)</math> be the area of <math>\triangle ABC</math>. Then the domain of <math>f</math> is an open interval <math>(p,q)</math>, and the maximum value <math>r</math> of <math>f(x)</math> occurs at <math>x=s</math>. What is <math>p+q+r+s</math>?
  
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<math>
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\textbf{(A) }909\qquad
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\textbf{(B) }910\qquad
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\textbf{(C) }911\qquad
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\textbf{(D) }912\qquad
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\textbf{(E) }913\qquad
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</math>

Revision as of 01:55, 14 November 2024

Problem 20

Suppose $A$, $B$, and $C$ are points in the plane with $AB=40$ and $AC=42$, and let $x$ be the length of the line segment from $A$ to the midpoint of $\overline{BC}$. Define a function $f$ by letting $f(x)$ be the area of $\triangle ABC$. Then the domain of $f$ is an open interval $(p,q)$, and the maximum value $r$ of $f(x)$ occurs at $x=s$. What is $p+q+r+s$?

$\textbf{(A) }909\qquad \textbf{(B) }910\qquad \textbf{(C) }911\qquad \textbf{(D) }912\qquad \textbf{(E) }913\qquad$

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