2023 AIME II Problems/Problem 12

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

Because $M$ is the midpoint of $BC$ , following from the Steward'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, $\frac{BQ}{\sin \angle BPA} = \frac{PQ}{\angle PBQ}$

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

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

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 (simplest)

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

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

~bluesoul

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 $\frac{99}{\sqrt{148}}$ —-> $99+148=\boxed{247}$

-dragoon

Video Solution 1 by SpreadTheMathLove

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

2023 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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