AIME 2010 II · 第 15 题

Solution 1 (Linearity)

Define the function $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ by

$$f(X)=\text{Pow}_{(AMN)}(X)-\text{Pow}_{(ADE)}(X)$$ for points $X$ in the plane. Then $f$ is linear, so $\frac{BQ}{CQ}=\frac{f(B)-f(Q)}{f(Q)-f(C)}$. But $f(Q)=0$ since $Q$ lies on the radical axis of $(AMN)$, $(ADE)$ thus

$$\frac{BQ}{CQ}=-\frac{f(B)}{f(C)}=-\frac{BN\cdot BA-BE\cdot BA}{CM\cdot CA-CD\cdot CA}$$ Let $AC=b$, $BC=a$ and $AB=c$. Note that $BN=\tfrac{c}{2}$ and $CM=\tfrac{b}{2}$ because they are midpoints, while $BE=\frac{ac}{a+b}$ and $CD=\frac{ab}{a+c}$ by Angle Bisector Theorem. Thus we can rewrite this expression as

$$\begin{aligned}&-\frac{\tfrac{c^{2}}{2}-\tfrac{ac^{2}}{a+b}}{\tfrac{b^{2}}{2}-\tfrac{ab^{2}}{a+c}} \\ =&-\left(\frac{c^{2}}{b^{2}}\right)\left(\frac{\tfrac{1}{2}-\tfrac{a}{a+b}}{\tfrac{1}{2}-\tfrac{a}{a+c}}\right) \\ =&\left(\frac{c^{2}}{b^{2}}\right)\left(\frac{a+c}{a+b}\right) \\ =&\left(\frac{225}{169}\right)\left(\frac{29}{27}\right) \\ =&~\frac{725}{507}\end{aligned}$$ so $m-n=\boxed{218}$.

Official Solution (MAA)

The Angle Bisector Theorem implies that $E$ lies on $\overline{AN}$ and $D$ lies on $\overline{MC}$ because $AE/EB = AC/BC < 1$ and $AD/DC = AB/CB > 1$. The Angle Bisector Theorem furthermore implies

$$NE=AN-AE=\frac 12\cdot AB - \frac{AC}{AC+BC}\cdot AB=\frac 5{18}$$ and

$$MD = CM - CD = \frac 12\cdot AC - \frac{BC}{BC+BA}\cdot AC = \frac{13}{58}.$$ Because $ANPM$ is cyclic, $\angle ENP = \angle ANP = \angle PMD$. Because $AEPD$ is cyclic, $\angle NEP = 180^\circ-\angle AEP = \angle MDP$. Because $\angle ENP =\angle PMD$ and $\angle NEP = \angle MDP$, triangles $NEP$ and $MDP$ are similar. Hence

$$\frac{NE}{MD}=\frac{NP}{MP}.$$ Applying the Law of Sines to $\triangle ANP$ and $\triangle AMP$ gives

$$\frac{NE}{MD} = \frac{NP}{MP} = \frac{\sin \angle NAP}{\sin \angle PAM} = \frac{\sin \angle BAQ}{\sin \angle QAC}$$ and thus

$$\frac{\sin \angle BAQ}{\sin \angle QAC} = \frac{(\frac{5}{18})}{(\frac{13}{58})} = \frac{145}{117}.$$ Thus

$$\frac{BQ}{QC} = \frac{\textrm{Area}(ABQ)}{\textrm{Area}(ACQ)} = \frac{\frac{1}{2}\cdot AB \cdot AQ \cdot \sin \angle BAQ}{\frac{1}{2}\cdot AC \cdot AQ \cdot \sin \angle QAC}= \frac{15}{13} \cdot \frac{145}{117} = \frac{725}{507},$$ and $m - n = 218$.

Solution 2

Let $Y = MN \cap AQ$. $\frac {BQ}{QC} = \frac {NY}{MY}$ since $\triangle AMN \sim \triangle ACB$. Since quadrilateral $AMPN$ is cyclic, $\triangle MYA \sim \triangle PYN$ and $\triangle MYP \sim \triangle AYN$, yielding $\frac {YM}{YA} = \frac {MP}{AN}$ and $\frac {YA}{YN} = \frac {AM}{PN}$. Multiplying these together yields *$\frac {YN}{YM} = \left(\frac {AN}{AM}\right) \left(\frac {PN}{PM}\right)$.

$\frac {AN}{AM} = \frac {\frac {AB}{2}}{\frac {AC}{2}} = \frac {15}{13}$.

Now we claim that $\triangle PMD \sim \triangle PNE$. To prove this, we can use cyclic quadrilaterals.

From $AMPN$, $\angle PNY \cong \angle PAM$ and $\angle ANM \cong \angle APM$. So, $m\angle PNA = m\angle PNY + m\angle ANM = m\angle PAM + m\angle APM = 180-m\angle PMA$ and $\angle PNA \cong \angle PMD$.

From $ADPE$, $\angle PDE \cong \angle PAE$ and $\angle EDA \cong \angle EPA$. Thus, $m\angle MDP = m\angle PDE + m\angle EDA = m\angle PAE + m\angle EPA = 180-m\angle PEA$ and $\angle PDM \cong \angle PEN$.

Thus, from AA similarity, $\triangle PMD \sim \triangle PNE$.

Therefore, $\frac {PN}{PM} = \frac {NE}{MD}$, which can easily be computed by the angle bisector theorem to be $\frac {145}{117}$. It follows that *$\frac {BQ}{CQ} = \frac {15}{13} \cdot \frac {145}{117} = \frac {725}{507}$, giving us an answer of $725 - 507 = \boxed{218}$.

• These two ratios are the same thing and can also be derived from the Ratio Lemma.

Ratio Lemma :$\frac{BD}{DC} = \frac{AB}{AC} \cdot \frac{\sin \angle BAD}{\sin \angle CAD}$, for any cevian AD of a triangle ABC. For the sine ratios use Law of Sines on triangles APM and APN,

$$\frac{PM}{\sin \angle PAM}=\frac{AP}{\sin \angle AMP}=\frac{AP}{\sin \angle ANP}=\frac{PN}{\sin \angle PAN}$$ . The information needed to use the Ratio Lemma can be found from the similar triangle section above.

Source: [1] by Zhero

Extension

The work done in this problem leads to a nice extension of this problem:

Given a $\triangle ABC$ and points $A_1$, $A_2$, $B_1$, $B_2$, $C_1$, $C_2$, such that $A_1$, $A_2$ $\in BC$, $B_1$, $B_2$ $\in AC$, and $C_1$, $C_2$ $\in AB$, then let $\omega_1$ be the circumcircle of $\triangle AB_1C_1$ and $\omega_2$ be the circumcircle of $\triangle AB_2C_2$. Let $A'$ be the intersection point of $\omega_1$ and $\omega_2$ distinct from $A$. Define $B'$ and $C'$ similarly. Then $AA'$, $BB'$, and $CC'$ concur.

This can be proven using Ceva's theorem and the work done in this problem, which effectively allows us to compute the ratio that line $AA'$ divides the opposite side $BC$ into and similarly for the other two sides.

Solution 3

This problem can be solved with barycentric coordinates. Let triangle $ABC$ be the reference triangle with $A=(1,0,0)$, $B=(0,1,0)$, and $C=(0,0,1)$. Thus, $N=(1:1:0)$ and $M=(1:0:1)$. Using the Angle Bisector Theorem, we can deduce that $D=(14:0:15)$ and $E = (14:13:0)$. Plugging the coordinates for triangles $ANM$ and $AED$ into the circle formula, we deduce that the equation for triangle $ANM$ is $-a^2yz-b^2zx-c^2xy+(\frac{c^2}{2}y+\frac{b^2}{2}z)(x+y+z)=0$ and the equation for triangle $AED$ is $-a^2yz-b^2zx-c^2xy+(\frac{14c^2}{27}y+\frac{14b^2}{29}z)(x+y+z)=0$. Solving the system of equations, we get that $\frac{c^2y}{54}=\frac{b^2z}{58}$. This equation determines the radical axis of circles $ANM$ and $AED$, on which points $P$ and $Q$ lie. Thus, solving for $\frac{z}{y}$ gets the desired ratio of lengths, and $\frac{z}{y}=\frac{58c^2}{54b^2}$ and plugging in the lengths $b=13$ and $c=15$ gets $\frac{725}{507}$. From this we get the desired answer of $725-507=\boxed{218}$. -wertguk

Solution 4 (Fast)

Observe that $P$ is the center of spiral symmetry of segments $DM$ and $EN$.

Using the angle bisector theorem, we compute $DM = \frac{13}{58}$ and $EN = \frac{5}{18}$. Hence, it follows that the side-length ratio of triangles $DMP$ to $ENP$ is $\frac{117}{145}$ (note that the two triangles are similar by the spiral symmetry). This implies that the ratio of the height from $P$ to $AC$ to the height from $P$ to $AB$ is $\frac{117}{145}$, so we compute the area ratio $\frac{APC}{APB} = \frac{507}{725}$.

From the above, we see that the barycentric coordinates of $P$ are of the form $(x : 507 : 725)$. Hence, it follows that the point $Q$ has the coordinates $(0 : 507 : 725)$, so $\frac{BQ}{CQ} = \frac{725}{507}$ and our answer is $\boxed{218}$.

~hgomamogh

Video Solution by the SpreadTheMathLove

https://www.youtube.com/watch?v=gzmY6nbbphw