Difference between revisions of "2023 AIME II Problems/Problem 12"

Latest revision as of 21:45, 19 August 2024

Problem

In $\triangle ABC$ with side lengths $AB = 13,$ $BC = 14,$ and $CA = 15,$ let $M$ be the midpoint of $\overline{BC}.$ Let $P$ be the point on the circumcircle of $\triangle ABC$ such that $M$ is on $\overline{AP}.$ There exists a unique point $Q$ on segment $\overline{AM}$ such that $\angle PBQ = \angle PCQ.$ Then $AQ$ can be written as $\frac{m}{\sqrt{n}},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution 1

Because $M$ is the midpoint of $BC$ , following from the Stewart's theorem, $AM = 2 \sqrt{37}$ .

Because $A$ , $B$ , $C$ , and $P$ are concyclic, $\angle BPA = \angle C$ , $\angle CPA = \angle B$ .

Denote $\theta = \angle PBQ$ .

In $\triangle BPQ$ , following from the law of sines, $\frac{BQ}{\sin \angle BPA} = \frac{PQ}{\sin \angle PBQ}$

Thus, $\frac{BQ}{\sin C} = \frac{PQ}{\sin \theta} . \hspace{1cm} (1)$

In $\triangle CPQ$ , following from the law of sines, $\frac{CQ}{\sin \angle CPA} = \frac{PQ}{\sin \angle PCQ}$

Thus, $\frac{CQ}{\sin B} = \frac{PQ}{\sin \theta} . \hspace{1cm} (2)$

Taking $\frac{(1)}{(2)}$ , we get $\frac{BQ}{\sin C} = \frac{CQ}{\sin B}$

In $\triangle ABC$ , following from the law of sines, $\frac{AB}{\sin C} = \frac{AC}{\sin B} . \hspace{1cm} (3)$

Thus, Equations (2) and (3) imply $\begin{align*} \frac{BQ}{CQ} & = \frac{AB}{AC} \\ & = \frac{13}{15} . \hspace{1cm} (4) \end{align*}$

Next, we compute $BQ$ and $CQ$ .

We have $\begin{align*} BQ^2 & = AB^2 + AQ^2 - 2 AB\cdot AQ \cos \angle BAQ \\ & = AB^2 + AQ^2 - 2 AB\cdot AQ \cos \angle BAM \\ & = AB^2 + AQ^2 - 2 AB\cdot AQ \cdot \frac{AB^2 + AM^2 - BM^2}{2 AB \cdot AM} \\ & = AB^2 + AQ^2 - AQ \cdot \frac{AB^2 + AM^2 - BM^2}{AM} \\ & = 169 + AQ^2 - \frac{268}{2 \sqrt{37}} AQ . \hspace{1cm} (5) \end{align*}$

We have $\begin{align*} CQ^2 & = AC^2 + AQ^2 - 2 AC\cdot AQ \cos \angle CAQ \\ & = AC^2 + AQ^2 - 2 AC\cdot AQ \cos \angle CAM \\ & = AC^2 + AQ^2 - 2 AC\cdot AQ \cdot \frac{AC^2 + AM^2 - CM^2}{2 AC \cdot AM} \\ & = AC^2 + AQ^2 - AQ \cdot \frac{AC^2 + AM^2 - CM^2}{AM} \\ & = 225 + AQ^2 - \frac{324}{2 \sqrt{37}} AQ . \hspace{1cm} (6) \end{align*}$

Taking (5) and (6) into (4), we get $AQ = \frac{99}{\sqrt{148}}$

Therefore, the answer is $99 + 148 = \boxed{\textbf{(247) }}$

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

Solution 2

Define $L_1$ to be the foot of the altitude from $A$ to $BC$ . Furthermore, define $L_2$ to be the foot of the altitude from $Q$ to $BC$ . From here, one can find $AL_1=12$ , either using the 13-14-15 triangle or by calculating the area of $ABC$ two ways. Then, we find $BL_1=5$ and $L_1C = 9$ using Pythagorean theorem. Let $QL_2=x$ . By AA similarity, $\triangle{AL_1M}$ and $\triangle{QL_2M}$ are similar. By similarity ratios, $\frac{AL_1}{L_1M}=\frac{QL_2}{L_2M}$ $\frac{12}{2}=\frac{x}{L_2M}$ $L_2M = \frac{x}{6}$ Thus, $BL_2=BM-L_2M=7-\frac{x}{6}$ . Similarly, $CL_2=7+\frac{x}{6}$ . Now, we angle chase from our requirement to obtain new information. $\angle{PBQ}=\angle{PCQ}$ $\angle{QCM}+\angle{PCM}=\angle{QBM}+\angle{PBM}$ $\angle{QCL_2}+\angle{PCM}=\angle{QBL_2}+\angle{PBM}$ $\angle{PCM}-\angle{PBM}=\angle{QBL_2}-\angle{QCL_2}$ $\angle{MAB}-\angle{MAC}=\angle{QBL_2}-\angle{QCL_2}\text{(By inscribed angle theorem)}$ $(\angle{MAL_1}+\angle{L_1AB})-(\angle{CAL_1}-\angle{MAL_1})=\angle{QBL_2}-\angle{QCL_2}$ $2\angle{MAL_1}+\angle{L_1AB}-\angle{CAL_1}=\angle{QBL_2}-\angle{QCL_2}$ Take the tangent of both sides to obtain $\tan(2\angle{MAL_1}+\angle{L_1AB}-\angle{CAL_1})=\tan(\angle{QBL_2}-\angle{QCL_2})$ By the definition of the tangent function on right triangles, we have $\tan{MAL_1}=\frac{7-5}{12}=\frac{1}{6}$ , $\tan{CAL_1}=\frac{9}{12}=\frac{3}{4}$ , and $\tan{L_1AB}=\frac{5}{12}$ . By abusing the tangent angle addition formula, we can find that $\tan(2\angle{MAL_1}+\angle{L_1AB}-\angle{CAL_1})=\frac{196}{2397}$ By substituting $\tan{\angle{QBL_2}}=\frac{6x}{42-x}$ , $\tan{\angle{QCL_2}}=\frac{6x}{42+x}$ and using tangent angle subtraction formula we find that $x=\frac{147}{37}$ Finally, using similarity formulas, we can find $\frac{AQ}{AM}=\frac{12-x}{x}$ . Plugging in $x=\frac{147}{37}$ and $AM=\sqrt{148}$ , we find that $AQ=\frac{99}{\sqrt{148}}$ Thus, our final answer is $99+148=\boxed{247}$ . ~sigma

Solution 3

It is clear that $BQCP$ is a parallelogram. By Stewart's Theorem, $AM=\sqrt{148}$ , POP on $M$ tells $PM=\frac{49}{\sqrt{148}}$

As $QM=PM, AQ=AM-PM=\frac{99}{\sqrt{148}}$ leads to $\boxed{247}$

~bluesoul (supplemental note: ~Mathavi)

$\textbf{NOTE: Why BQCP is a parallelogram}$

It's not actually immediately clear why this is the case. There are two ways to easily show this:

$\textbf{Competition solution:}$

Notice that the problem statement tells us that point Q is $\textit{unique.}$ EVERY piece of information in the problem statement is intentional, so we should try to use this to our benefit. None of the other solutions do, which is why they are more complicated than they need be.

Consider point Q' s.t. $Q'M = MP$ . Obviously, $\angle Q'CP$ and $\angle Q'BP$ are equal - we have perfect symmetry along line $AP$ . Moreover, $BQ'CP$ is a parallelogram as its diagonals bisect each other. Since point $Q$ is unique, we know that $Q' \textit{is } Q$ . Thus $BQCP$ is a parallelogram. $\blacksquare$ $\newline$ $\textbf{Rigorous proof (not recommended for competition scenario):}$ Consider any quadrilateral $ABCD$ whose diagonals intersect at $O$ s.t. $AO = OC$ and $\angle BAD = \angle BCD$ . We will prove that $ABCD$ is $\textit{either a \textbf{parallelogram} or a \textbf{kite}}$ .

(Note that in our problem, since $AP$ and $BC$ are not orthogonal, ( $ABC$ isn't isosceles) this is enough to show that $BQCP$ is a parallelogram). $\newline$ -- By same base/same altitude, $[ABO] = [CBO]$ and $[ADO] = [CDO] \implies [ABD] = [ABO] + [ADO] = [CBO] + [CDO] = [CBD]. \newline$

Therefore: $\frac{1}{2} sin(\angle BAD) \overline{AB} \times \overline{AD} = \frac{1}{2} sin(\angle BCD) \overline{CB} \times \overline{CD}.$ Since $\angle BAD = \angle BCD$ , this reduces to $\overline{AB} \times \overline{AD} = \overline{CB} \times \overline{CD}. (E.1) \newline$

Let $AB = x$ and $AD = y$ . Then, by $(E.1)$ , $CB = kx$ and $CD = \frac{y}{k}$ for some real $k$ . Then by LoC on $\triangle BAD$ and $\triangle BCD$ : $x^{2} + y^{2} - 2xy cos(\angle BAD) = \overline{BD} = x^{2}k^{2} + \frac{y^{2}}{k^{2}} - 2xy cos(\angle BCD) \newline \implies x^{2} + y^{2} = x^{2}k^{2} + \frac{y^{2}}{k^{2}} \newline \implies (y^{2} - x^{2}k^{2})(k^{2} - 1) = 0.\newline$

-- $y^{2} - x^{2}k^{2} = 0 \implies y = kx \implies AD = BC$ and $AB = CD \implies$ $ABCD$ is a parallelogram.

-- $k^{2} - 1 = 0 \implies k = 1$ ( $k$ cannot be $-1$ ; no negative sided polygons here!) $\implies AB = CB$ and $AD = CD \implies$ $ABCD$ is a kite. $\square$ . ~Mathavi

Solution 4 (LOS+ coordbash)

First, note that by Law of Sines, $\frac{\sin(\angle{QBP})}{QP}=\frac{\sin(\angle{BPQ})}{BQ}$ and that $\frac{\sin(\angle{QCP})}{QP}=\frac{\sin(\angle{QPC})}{QP}$ . Equating the 2 expressions, you get that $\frac{\sin(\angle{BPQ})}{BQ}=\frac{\sin(\angle{QPC})}{QP}$ . Now drop the altitude from $A$ to $BC$ . As it is commonly known that the dropped altitude forms a $5-12-13$ and a $9-12-15$ triangle, you get the measures of $\angle{ABC}$ and $\angle{ACB}$ respectively, which are $\arcsin(\frac{12}{13})$ and $\arcsin(\frac{4}{5})$ . However, by the inscribed angle theorem, you get that $\angle{BPQ}=\arcsin(\frac{4}{5})$ and that $\angle{QPC}=\arcsin(\frac{12}{13})$ , respectively. Therefore, by Law of Sines (as previously stated) $\frac{BQ}{CQ}=\frac{13}{15}$ .

Now commence coordbashing. Let $B$ be the origin, and $A$ be the point $(5,12)$ . As $AP$ passes through $A$ , which is $(5,12)$ , and $M$ , which is $(7,0)$ , it has the equation $-6x+42$ , so therefore a point on this line can be written as $(x,42-6x)$ . As we have the ratio of the lengths, which prompts us to write the lengths in terms of the distance formula, we can just plug and chug it in to get the ratio $\frac{\sqrt{37x^2-504x+1764}}{\sqrt{37x^2-532x+1960}}=\frac{13}{15}$ . This can be squared to get $\frac{37x^2-504x+1764}{37x^2-532x+1960}=\frac{169}{225}$ . This can be solved to get a solution of $x=\frac{469}{74}$ , and an extraneous solution of $5$ which obviously doesn’t work.

Plugging $x$ into the line equation gets you $y=\frac{147}{37}$ . The distance between this point and $A$ , which is $(5,12)$ is $\sqrt{\frac{9801}{148}}$ , or simplified to $\frac{99}{\sqrt{148}}\Longrightarrow99+148=\boxed{247}$

~dragoon (minor $\LaTeX$ fixes by rhydon516)

Solution 5 (similar to 3)

We use the law of Cosine and get $AB^2 = AM^2 + BM^2 - 2 AM \cdot BM \cos \angle AMB,$ $AC^2 = AM^2 + CM^2 + 2 AM \cdot CM \cos \angle AMB \implies$ $AM^2 = \frac {AB^2 + AC^2}{2}- BM^2 = \sqrt{148} \approx 12.$ We use the power of point $M$ with respect circumcircle $\triangle ABC$ and get $AM \cdot MP = BM \cdot CM = BM^2 \implies$ $PM = \frac {49}{\sqrt {148}} \approx \frac {48}{12} \approx 4 < AM.$ It is clear that if $Q = P,$ then $\angle PBQ = \angle PCQ = 0 \implies$

if $Q$ is symmetric to $P$ with respect $M$ then $\angle PBQ = \angle PCQ.$

There exists a unique point $Q$ on segment $\overline{AM}, PM < AM \implies$ $PQ = AM - PM = \frac{99}{\sqrt{148}} \implies \boxed{247}.$ vladimir.shelomovskii@gmail.com, vvsss

Video Solution 1 (Fastest and Easiest)

https://www.youtube.com/watch?v=qm9Sg1tEJJE

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=k6hEFEVVzMI

@@ Line 3: / Line 3: @@
 In <math>\triangle ABC</math> with side lengths <math>AB = 13,</math> <math>BC = 14,</math> and <math>CA = 15,</math> let <math>M</math> be the midpoint of <math>\overline{BC}.</math> Let <math>P</math> be the point on the circumcircle of <math>\triangle ABC</math> such that <math>M</math> is on <math>\overline{AP}.</math> There exists a unique point <math>Q</math> on segment <math>\overline{AM}</math> such that <math>\angle PBQ = \angle PCQ.</math> Then <math>AQ</math> can be written as <math>\frac{m}{\sqrt{n}},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math>
-==Solution==
+==Solution 1==
-Because <math>M</math> is the midpoint of <math>BC</math>, following from the Steward's theorem, <math>AM = 2 \sqrt{37}</math>.
+Because <math>M</math> is the midpoint of <math>BC</math>, following from the Stewart's theorem, <math>AM = 2 \sqrt{37}</math>.
-Because <math>A</math>, <math>B</math>, <math>C</math>, <math>P</math> are concyclic, <math>\angle BPA = \angle C</math>, <math>\angle CPA = \angle B</math>.
+Because <math>A</math>, <math>B</math>, <math>C</math>, and <math>P</math> are concyclic, <math>\angle BPA = \angle C</math>, <math>\angle CPA = \angle B</math>.
 Denote <math>\theta = \angle PBQ</math>.
@@ Line 14: / Line 14: @@
 <cmath>
 \[
-\frac{BQ}{\sin \angle BPA} = \frac{PQ}{\angle PBQ} .
+\frac{BQ}{\sin \angle BPA} = \frac{PQ}{\sin \angle PBQ}
 \]
 </cmath>
@@ Line 28: / Line 28: @@
 <cmath>
 \[
-\frac{CQ}{\sin \angle CPA} = \frac{PQ}{\angle PCQ} .
+\frac{CQ}{\sin \angle CPA} = \frac{PQ}{\sin \angle PCQ}
 \]
 </cmath>
@@ Line 42: / Line 42: @@
 <cmath>
 \[
-\frac{BQ}{\sin C} = \frac{CQ}{\sin B} .
+\frac{BQ}{\sin C} = \frac{CQ}{\sin B}
 \]
 </cmath>
@@ Line 87: / Line 87: @@
 </cmath>
-Taking (5) and (6) into (4), we get <math>AQ = \frac{99}{\sqrt{148}}</math>.
+Taking (5) and (6) into (4), we get <math>AQ = \frac{99}{\sqrt{148}}</math>
 Therefore, the answer is <math>99 + 148 = \boxed{\textbf{(247) }}</math>
@@ Line 114: / Line 115: @@
 ~sigma
-==Solution 3 (simplest)==
+==Solution 3==
-It is clear that <math>BQCP</math> is a parallelogram. By Stewart's Theorem, <math>AM=\sqrt{148}</math>, POP tells <math>PM=\frac{49}{\sqrt{148}}</math>
+It is clear that <math>BQCP</math> is a parallelogram. By Stewart's Theorem, <math>AM=\sqrt{148}</math>, POP on <math>M</math> tells <math>PM=\frac{49}{\sqrt{148}}</math>
 As <math>QM=PM, AQ=AM-PM=\frac{99}{\sqrt{148}}</math> leads to <math>\boxed{247}</math>
-~bluesoul
+~bluesoul (supplemental note: ~Mathavi)
+<math>\textbf{NOTE: Why BQCP is a parallelogram}</math>
+It's not actually immediately clear why this is the case. There are two ways to easily show this:
+<math>\textbf{Competition solution:}</math>
+Notice that the problem statement tells us that point Q is <math>\textit{unique.}</math> EVERY piece of information in the problem statement is intentional, so we should try to use this to our benefit. None of the other solutions do, which is why they are more complicated than they need be.
+Consider point Q' s.t. <math>Q'M = MP</math>. Obviously, <math>\angle Q'CP</math> and <math>\angle Q'BP</math> are equal - we have perfect symmetry along line <math>AP</math>. Moreover, <math>BQ'CP</math> is a parallelogram as its diagonals bisect each other. Since point <math>Q</math> is unique, we know that <math>Q' \textit{is } Q</math>. Thus <math>BQCP</math> is a parallelogram. <math>\blacksquare</math>
+<math>\newline</math>
+<math>\textbf{Rigorous proof (not recommended for competition scenario):}</math>
+Consider any quadrilateral <math>ABCD</math> whose diagonals intersect at <math>O</math> s.t. <math>AO = OC</math> and <math>\angle BAD = \angle BCD</math>. We will prove that <math>ABCD</math> is <math>\textit{either a \textbf{parallelogram} or a \textbf{kite}}</math>.
+(Note that in our problem, since <math>AP</math> and <math>BC</math> are not orthogonal, (<math>ABC</math> isn't isosceles) this is enough to show that <math>BQCP</math> is a parallelogram).
+<math>\newline</math>
+-- By same base/same altitude, <math>[ABO] = [CBO]</math> and <math>[ADO] = [CDO] \implies [ABD] = [ABO] + [ADO] = [CBO] + [CDO] = [CBD]. \newline</math>
+Therefore: <math>\frac{1}{2} sin(\angle BAD) \overline{AB} \times \overline{AD} = \frac{1}{2} sin(\angle BCD) \overline{CB} \times \overline{CD}.</math> Since <math>\angle BAD = \angle BCD</math>, this reduces to <math>\overline{AB} \times \overline{AD} = \overline{CB} \times \overline{CD}. (E.1) \newline</math>
+Let <math>AB = x</math> and <math>AD = y</math>. Then, by <math>(E.1)</math>, <math>CB = kx</math> and <math>CD = \frac{y}{k}</math> for some real <math>k</math>. Then by LoC on <math>\triangle BAD</math> and <math>\triangle BCD</math>:
+<math>x^{2} + y^{2} - 2xy cos(\angle BAD) = \overline{BD} = x^{2}k^{2} + \frac{y^{2}}{k^{2}} - 2xy cos(\angle BCD) \newline \implies x^{2} + y^{2} = x^{2}k^{2} + \frac{y^{2}}{k^{2}} \newline \implies (y^{2} - x^{2}k^{2})(k^{2} - 1) = 0.\newline</math>
+-- <math>y^{2} - x^{2}k^{2} = 0 \implies y = kx \implies AD = BC</math> and <math>AB = CD \implies</math> <math>ABCD</math> is a parallelogram.
+-- <math>k^{2} - 1 = 0 \implies k = 1</math> (<math>k</math> cannot be <math>-1</math>; no negative sided polygons here!) <math>\implies AB = CB</math> and <math>AD = CD \implies</math> <math>ABCD</math> is a kite. <math>\square</math>. ~Mathavi
+==Solution 4 (LOS+ coordbash)==
+First, note that by Law of Sines, <math>\frac{\sin(\angle{QBP})}{QP}=\frac{\sin(\angle{BPQ})}{BQ}</math> and that <math>\frac{\sin(\angle{QCP})}{QP}=\frac{\sin(\angle{QPC})}{QP}</math>. Equating the 2 expressions, you get that <math>\frac{\sin(\angle{BPQ})}{BQ}=\frac{\sin(\angle{QPC})}{QP}</math>. Now drop the altitude from <math>A</math> to <math>BC</math>. As it is commonly known that the dropped altitude forms a <math>5-12-13</math> and a <math>9-12-15</math> triangle, you get the measures of <math>\angle{ABC}</math> and <math>\angle{ACB}</math> respectively, which are <math>\arcsin(\frac{12}{13})</math> and <math>\arcsin(\frac{4}{5})</math>. However, by the inscribed angle theorem, you get that <math>\angle{BPQ}=\arcsin(\frac{4}{5})</math> and that <math>\angle{QPC}=\arcsin(\frac{12}{13})</math>, respectively. Therefore, by Law of Sines (as previously stated) <math>\frac{BQ}{CQ}=\frac{13}{15}</math>.
+Now commence coordbashing. Let <math>B</math> be the origin, and <math>A</math> be the point <math>(5,12)</math>. As <math>AP</math> passes through <math>A</math>, which is <math>(5,12)</math>, and <math>M</math>, which is <math>(7,0)</math>, it has the equation <math>-6x+42</math>, so therefore a point on this line can be written as <math>(x,42-6x)</math>. As we have the ratio of the lengths, which prompts us to write the lengths in terms of the distance formula, we can just plug and chug it in to get the ratio <math>\frac{\sqrt{37x^2-504x+1764}}{\sqrt{37x^2-532x+1960}}=\frac{13}{15}</math>. This can be squared to get <math>\frac{37x^2-504x+1764}{37x^2-532x+1960}=\frac{169}{225}</math>. This can be solved to get a solution of <math>x=\frac{469}{74}</math>, and an extraneous solution of <math>5</math> which obviously doesn’t work.
+Plugging <math>x</math> into the line equation gets you <math>y=\frac{147}{37}</math>. The distance between this point and <math>A</math>, which is <math>(5,12)</math> is <math>\sqrt{\frac{9801}{148}}</math>, or simplified to <math>\frac{99}{\sqrt{148}}\Longrightarrow99+148=\boxed{247}</math>
+~dragoon (minor <math>\LaTeX</math> fixes by rhydon516)
+==Solution 5 (similar to 3)==
+[[File:2023 AIME II 12.png|270px|right]]
+We use the law of Cosine and get <cmath>AB^2 = AM^2 + BM^2 - 2 AM \cdot BM \cos \angle AMB,</cmath> <cmath>AC^2 = AM^2 + CM^2 + 2 AM \cdot CM \cos \angle AMB \implies</cmath> <cmath>AM^2 = \frac {AB^2 + AC^2}{2}- BM^2 = \sqrt{148} \approx 12.</cmath>
+We use the power of point <math>M</math> with respect circumcircle <math>\triangle ABC</math> and get
+<cmath>AM \cdot MP = BM \cdot CM = BM^2 \implies</cmath>
+<cmath>PM = \frac {49}{\sqrt {148}} \approx \frac {48}{12} \approx 4 < AM.</cmath>
+It is clear that if <math>Q = P,</math> then <math>\angle PBQ = \angle PCQ = 0 \implies</math>
+if <math>Q</math> is symmetric to <math>P</math> with respect <math>M</math> then <math>\angle PBQ = \angle PCQ.</math>
+There exists a <i><b>unique</b></i> point <math>Q</math> on segment <math>\overline{AM}, PM < AM \implies</math>
+<cmath>PQ = AM - PM = \frac{99}{\sqrt{148}} \implies \boxed{247}.</cmath>
+'''vladimir.shelomovskii@gmail.com, vvsss'''
+==Video Solution 1 (Fastest and Easiest) ==
+https://www.youtube.com/watch?v=qm9Sg1tEJJE
+==Video Solution 2 by SpreadTheMathLove==
+https://www.youtube.com/watch?v=k6hEFEVVzMI
 == See also ==
 {{AIME box|year=2023|num-b=11|num-a=13|n=II}}
+[[Category:Intermediate Geometry Problems]]
 {{MAA Notice}}

2023 AIME II (Problems • Answer Key • Resources)
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