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Difference between revisions of "2024 AIME I Problems/Problem 10"

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==Problem==
 
==Problem==
−
Let <math>ABC</math> be a triangle inscribed in circle <math>\omega</math>. Let the tangents to <math>\omega</math> at <math>B</math> and <math>C</math> intersect at point <math>P</math>, and let <math>\overline{AP}</math> intersect <math>\omega</math> at <math>D</math>. Find <math>AD</math>, if <math>AB=5</math>, <math>BC=9</math>, and <math>AC=10</math>.
+
Let <math>ABC</math> be a triangle inscribed in circle <math>\omega</math>. Let the tangents to <math>\omega</math> at <math>B</math> and <math>C</math> intersect at point <math>D</math>, and let <math>\overline{AD}</math> intersect <math>\omega</math> at <math>P</math>. If <math>AB=5</math>, <math>BC=9</math>, and <math>AC=10</math>, <math>AP</math> can be written as the form <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime integers. Find <math>m + n</math>.
 +
 
 +
==Solution 1==
 +
From the tangency condition we have <math>\let\angle BCD = \let\angle CBD = \let\angle A</math>. With LoC we have <math>\cos(A) = \frac{25+100-81}{2*5*10} = \frac{11}{25}</math> and <math>\cos(B) = \frac{81+25-100}{2*9*5} = \frac{1}{15}</math>. Then, <math>CD = \frac{\frac{9}{2}}{\cos(A)} = \frac{225}{22}</math>. Using LoC we can find <math>AD</math>: <math>AD^2 = AC^2 + CD^2 - 2(AC)(CD)\cos(A+C) = 10^2+(\frac{225}{22})^2 + 2(10)\frac{225}{22}\cos(B) = 100 + \frac{225^2}{22^2} + 2(10)\frac{225}{22}*\frac{1}{15} = \frac{5^4*13^2}{484}</math>. Thus, <math>AD = \frac{5^2*13}{22}</math>. By Power of a Point, <math>DP*AD = CD^2</math> so <math>DP*\frac{5^2*13}{22} = (\frac{225}{22})^2</math> which gives <math>DP = \frac{5^2*9^2}{13*22}</math>. Finally, we have <math>AP = AD - DP = \frac{5^2*13}{22} - \frac{5^2*9^2}{13*22} = \frac{100}{13} \rightarrow \boxed{113}</math>.
 +
 
 +
~angie.
 +
 
 +
==Solution 2==
 +
 
 +
We know <math>AP</math> is the symmedian, which implies <math>\triangle{ABP}\sim \triangle{AMC}</math> where <math>M</math> is the midpoint of <math>BC</math>. By Appolonius theorem, <math>AM=\frac{13}{2}</math>. Thus, we have <math>\frac{AP}{AC}=\frac{AB}{AM}, AP=\frac{100}{13}\implies \boxed{113}</math>
 +
 
 +
~Bluesoul
 +
 
 +
==Solution 3==
 +
 
 +
Extend sides <math>\overline{AB}</math> and <math>\overline{AC}</math> to points <math>E</math> and <math>F</math>, respectively, such that <math>B</math> and <math>C</math> are the feet of the altitudes in <math>\triangle AEF</math>. Denote the feet of the altitude from <math>A</math> to <math>\overline{EF}</math> as <math>X</math>, and let <math>H</math> denote the orthocenter of <math>\triangle AEF</math>. Call <math>M</math> the midpoint of segment <math>\overline{EF}</math>. By the Three Tangents Lemma, we have that <math>MB</math> and <math>MC</math> are both tangents to <math>(ABC)</math> <math>\implies</math> <math>M = D</math>, and since <math>M</math> is the midpoint of <math>\overline{EF}</math>, <math>MF = MB</math>. Additionally, by angle chasing, we get that:
 +
<cmath>\angle ABC \cong \angle AHC \cong \angle EHX</cmath>
 +
Also,
 +
<cmath>\angle EHX = 90 ^\circ - \angle HEF = 90 ^\circ - (90 ^\circ - \angle AFE) = \angle AFE</cmath>
 +
Furthermore,
 +
<cmath>AB = AF \cdot \cos(A)</cmath>
 +
From this, we see that <math>\triangle ABC \sim \triangle AFE</math> with a scale factor of <math>\cos(A)</math>. By the Law of Cosines,
 +
<cmath>\cos(A) = \frac{10^2 + 5^2 - 9^2}{2 \cdot 10 \cdot 5} = \frac{11}{25}</cmath>
 +
Thus, we can find that the side lengths of <math>\triangle AEF</math> are <math>\frac{250}{11}, \frac{125}{11}, \frac{225}{11}</math>. Then, by Stewart's theorem, <math>AM = \frac{13 \cdot 25}{22}</math>. By Power of a Point,
 +
<cmath>\overline{MB} \cdot \overline{MB} = \overline{MA} \cdot \overline{MP}</cmath>
 +
<cmath>\frac{225}{22} \cdot \frac{225}{22} = \overline{MP} \cdot \frac{13 \cdot 25}{22} \implies \overline{MP} = \frac{225 \cdot 9}{22 \cdot 13}</cmath>
 +
Thus,
 +
<cmath>AP = AM - MP = \frac{13 \cdot 25}{22} - \frac{225 \cdot 9}{22 \cdot 13} = \frac{100}{13}</cmath>
 +
Therefore, the answer is <math>\boxed{113}</math>.
 +
 
 +
~mathwiz_1207
 +
 
 +
==Solution 4 (LoC spam)==
 +
Connect lines <math>\overline{PB}</math> and <math>\overline{PC}</math>. From the angle by tanget formula, we have <math>\angle PBD = \angle DAB</math>. Therefore by AA similarity, <math>\triangle PBD \sim \triangle BAD</math>. Let <math>\overline{BP} = x</math>. Using ratios, we have <cmath>\frac{x}{5}=\frac{BD}{AD}.</cmath> Similarly, using angle by tangent, we have <math>\angle PCD = \angle DAC</math>, and by AA similarity, <math>\triangle CPD \sim \triangle ACD</math>. By ratios, we have <cmath>\frac{PC}{10}=\frac{CD}{AD}.</cmath> However, because <math>\overline{BD}=\overline{CD}</math>, we have <cmath>\frac{x}{5}=\frac{PC}{10},</cmath> so <math>\overline{PC}=2x.</math> Now using Law of Cosines on <math>\angle BAC</math> in triangle <math>\triangle ABC</math>, we have <cmath>9^2=5^2+10^2-100\cos(\angle BAC).</cmath> Solving, we find <math>\cos(\angle BAC)=\frac{11}{25}</math>. Now we can solve for <math>x</math>. Using Law of Cosines on <math>\triangle BPC,</math> we have
 +
\begin{align*}
 +
81&=x^2+4x^2-4x^2\cos(180-\angle BAC) \\
 +
&= 5x^2+4x^2\cos(BAC). \\
 +
\end{align*}
 +
 
 +
Solving, we get <math>x=\frac{45}{13}.</math> Now we have a system of equations using Law of Cosines on <math>\triangle BPA</math> and <math>\triangle CPA</math>, <cmath>AP^2=5^2+\left(\frac{45}{13}\right)^2 -(10) \left(\frac{45}{13} \right)\cos(ABP)</cmath>
 +
<cmath>AP^2=10^2+4 \left(\frac{45}{13} \right)^2 + (40) \left(\frac{45}{13} \right)\cos(ABP).</cmath>
 +
 
 +
Solving, we find <math>\overline{AP}=\frac{100}{13}</math>, so our desired answer is <math>100+13=\boxed{113}</math>.
 +
 
 +
~evanhliu2009
 +
 
 +
==Solution 5==
 +
 
 +
Following from the law of cosines, we can easily get <math>\cos A = \frac{11}{25}</math>, <math>\cos B = \frac{1}{15}</math>, <math>\cos C = \frac{13}{15}</math>.
 +
 
 +
Hence, <math>\sin A = \frac{6 \sqrt{14}}{25}</math>, <math>\cos 2C = \frac{113}{225}</math>, <math>\sin 2C = \frac{52 \sqrt{14}}{225}</math>.
 +
Thus, <math>\cos \left( A + 2C \right) = - \frac{5}{9}</math>.
 +
 
 +
Denote by <math>R</math> the circumradius of <math>\triangle ABC</math>.
 +
In <math>\triangle ABC</math>, following from the law of sines, we have <math>R = \frac{BC}{2 \sin A} = \frac{75}{4 \sqrt{14}}</math>.
 +
 
 +
Because <math>BD</math> and <math>CD</math> are tangents to the circumcircle <math>ABC</math>, <math>\triangle OBD \cong \triangle OCD</math> and <math>\angle OBD = 90^\circ</math>.
 +
Thus, <math>OD = \frac{OB}{\cos \angle BOD} = \frac{R}{\cos A}</math>.
 +
 
 +
In <math>\triangle AOD</math>, we have <math>OA = R</math> and <math>\angle AOD = \angle BOD + \angle AOB = A + 2C</math>.
 +
Thus, following from the law of cosines, we have
 +
 
 +
\begin{align*}
 +
AD & = \sqrt{OA^2 + OD^2 - 2 OA \cdot OD \cos \angle AOD} \\
 +
& = \frac{26 \sqrt{14}}{33} R.
 +
\end{align*}
 +
 
 +
 
 +
Following from the law of cosines,
 +
 
 +
\begin{align*}
 +
\cos \angle OAD & = \frac{AD^2 + OA^2 - OD^2}{2 AD \cdot OA} \\
 +
& = \frac{8 \sqrt{14}}{39} .
 +
\end{align*}
 +
 
 +
 
 +
Therefore,
 +
 
 +
\begin{align*}
 +
AP & = 2 OA \cos \angle OAD \\
 +
& = \frac{100}{13} .
 +
\end{align*}
 +
 
 +
 
 +
Therefore, the answer is <math>100 + 13 = \boxed{\textbf{(113) }}</math>.
 +
 
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 +
 
 +
==Video Solution 1 by OmegaLearn.org==
 +
https://youtu.be/heryP002bp8
 +
 
 +
==Video Solution==
 +
 
 +
https://youtu.be/RawwQmVYyaw
 +
 
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
  
 
==See also==
 
==See also==

Latest revision as of 21:17, 6 February 2024

Problem

Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$.

Solution 1

From the tangency condition we have $\let\angle BCD = \let\angle CBD = \let\angle A$. With LoC we have $\cos(A) = \frac{25+100-81}{2*5*10} = \frac{11}{25}$ and $\cos(B) = \frac{81+25-100}{2*9*5} = \frac{1}{15}$. Then, $CD = \frac{\frac{9}{2}}{\cos(A)} = \frac{225}{22}$. Using LoC we can find $AD$: $AD^2 = AC^2 + CD^2 - 2(AC)(CD)\cos(A+C) = 10^2+(\frac{225}{22})^2 + 2(10)\frac{225}{22}\cos(B) = 100 + \frac{225^2}{22^2} + 2(10)\frac{225}{22}*\frac{1}{15} = \frac{5^4*13^2}{484}$. Thus, $AD = \frac{5^2*13}{22}$. By Power of a Point, $DP*AD = CD^2$ so $DP*\frac{5^2*13}{22} = (\frac{225}{22})^2$ which gives $DP = \frac{5^2*9^2}{13*22}$. Finally, we have $AP = AD - DP = \frac{5^2*13}{22} - \frac{5^2*9^2}{13*22} = \frac{100}{13} \rightarrow \boxed{113}$.

~angie.

Solution 2

We know $AP$ is the symmedian, which implies $\triangle{ABP}\sim \triangle{AMC}$ where $M$ is the midpoint of $BC$. By Appolonius theorem, $AM=\frac{13}{2}$. Thus, we have $\frac{AP}{AC}=\frac{AB}{AM}, AP=\frac{100}{13}\implies \boxed{113}$

~Bluesoul

Solution 3

Extend sides $\overline{AB}$ and $\overline{AC}$ to points $E$ and $F$, respectively, such that $B$ and $C$ are the feet of the altitudes in $\triangle AEF$. Denote the feet of the altitude from $A$ to $\overline{EF}$ as $X$, and let $H$ denote the orthocenter of $\triangle AEF$. Call $M$ the midpoint of segment $\overline{EF}$. By the Three Tangents Lemma, we have that $MB$ and $MC$ are both tangents to $(ABC)$ $\implies$ $M = D$, and since $M$ is the midpoint of $\overline{EF}$, $MF = MB$. Additionally, by angle chasing, we get that: \[\angle ABC \cong \angle AHC \cong \angle EHX\] Also, \[\angle EHX = 90 ^\circ - \angle HEF = 90 ^\circ - (90 ^\circ - \angle AFE) = \angle AFE\] Furthermore, \[AB = AF \cdot \cos(A)\] From this, we see that $\triangle ABC \sim \triangle AFE$ with a scale factor of $\cos(A)$. By the Law of Cosines, \[\cos(A) = \frac{10^2 + 5^2 - 9^2}{2 \cdot 10 \cdot 5} = \frac{11}{25}\] Thus, we can find that the side lengths of $\triangle AEF$ are $\frac{250}{11}, \frac{125}{11}, \frac{225}{11}$. Then, by Stewart's theorem, $AM = \frac{13 \cdot 25}{22}$. By Power of a Point, \[\overline{MB} \cdot \overline{MB} = \overline{MA} \cdot \overline{MP}\] \[\frac{225}{22} \cdot \frac{225}{22} = \overline{MP} \cdot \frac{13 \cdot 25}{22} \implies \overline{MP} = \frac{225 \cdot 9}{22 \cdot 13}\] Thus, \[AP = AM - MP = \frac{13 \cdot 25}{22} - \frac{225 \cdot 9}{22 \cdot 13} = \frac{100}{13}\] Therefore, the answer is $\boxed{113}$.

~mathwiz_1207

Solution 4 (LoC spam)

Connect lines $\overline{PB}$ and $\overline{PC}$. From the angle by tanget formula, we have $\angle PBD = \angle DAB$. Therefore by AA similarity, $\triangle PBD \sim \triangle BAD$. Let $\overline{BP} = x$. Using ratios, we have \[\frac{x}{5}=\frac{BD}{AD}.\] Similarly, using angle by tangent, we have $\angle PCD = \angle DAC$, and by AA similarity, $\triangle CPD \sim \triangle ACD$. By ratios, we have \[\frac{PC}{10}=\frac{CD}{AD}.\] However, because $\overline{BD}=\overline{CD}$, we have \[\frac{x}{5}=\frac{PC}{10},\] so $\overline{PC}=2x.$ Now using Law of Cosines on $\angle BAC$ in triangle $\triangle ABC$, we have \[9^2=5^2+10^2-100\cos(\angle BAC).\] Solving, we find $\cos(\angle BAC)=\frac{11}{25}$. Now we can solve for $x$. Using Law of Cosines on $\triangle BPC,$ we have \begin{align*} 81&=x^2+4x^2-4x^2\cos(180-\angle BAC) \\ &= 5x^2+4x^2\cos(BAC). \\ \end{align*}

Solving, we get $x=\frac{45}{13}.$ Now we have a system of equations using Law of Cosines on $\triangle BPA$ and $\triangle CPA$, \[AP^2=5^2+\left(\frac{45}{13}\right)^2 -(10) \left(\frac{45}{13} \right)\cos(ABP)\] \[AP^2=10^2+4 \left(\frac{45}{13} \right)^2 + (40) \left(\frac{45}{13} \right)\cos(ABP).\]

Solving, we find $\overline{AP}=\frac{100}{13}$, so our desired answer is $100+13=\boxed{113}$.

~evanhliu2009

Solution 5

Following from the law of cosines, we can easily get $\cos A = \frac{11}{25}$, $\cos B = \frac{1}{15}$, $\cos C = \frac{13}{15}$.

Hence, $\sin A = \frac{6 \sqrt{14}}{25}$, $\cos 2C = \frac{113}{225}$, $\sin 2C = \frac{52 \sqrt{14}}{225}$. Thus, $\cos \left( A + 2C \right) = - \frac{5}{9}$.

Denote by $R$ the circumradius of $\triangle ABC$. In $\triangle ABC$, following from the law of sines, we have $R = \frac{BC}{2 \sin A} = \frac{75}{4 \sqrt{14}}$.

Because $BD$ and $CD$ are tangents to the circumcircle $ABC$, $\triangle OBD \cong \triangle OCD$ and $\angle OBD = 90^\circ$. Thus, $OD = \frac{OB}{\cos \angle BOD} = \frac{R}{\cos A}$.

In $\triangle AOD$, we have $OA = R$ and $\angle AOD = \angle BOD + \angle AOB = A + 2C$. Thus, following from the law of cosines, we have

\begin{align*} AD & = \sqrt{OA^2 + OD^2 - 2 OA \cdot OD \cos \angle AOD} \\ & = \frac{26 \sqrt{14}}{33} R. \end{align*}


Following from the law of cosines,

\begin{align*} \cos \angle OAD & = \frac{AD^2 + OA^2 - OD^2}{2 AD \cdot OA} \\ & = \frac{8 \sqrt{14}}{39} . \end{align*}


Therefore,

\begin{align*} AP & = 2 OA \cos \angle OAD \\ & = \frac{100}{13} . \end{align*}


Therefore, the answer is $100 + 13 = \boxed{\textbf{(113) }}$.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution 1 by OmegaLearn.org

https://youtu.be/heryP002bp8

Video Solution

https://youtu.be/RawwQmVYyaw

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

See also

2024 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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All AIME Problems and Solutions

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