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Difference between revisions of "2015 AIME II Problems/Problem 11"

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==Problem==
 
==Problem==
  
−
The circumcircle of acute <math>\triangle ABC</math> has center <math>O</math>. The line passing through point <math>O</math> perpendicular to <math>\overline{OB}</math> intersects lines <math>AB</math> and <math>BC</math> and <math>P</math> and <math>Q</math>, respectively. Also <math>AB=5</math>, <math>BC=4</math>, <math>BQ=4.5</math>, and <math>BP=\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
+
The circumcircle of acute <math>\triangle ABC</math> has center <math>O</math>. The line passing through point <math>O</math> perpendicular to <math>\overline{OB}</math> intersects lines <math>AB</math> and <math>BC</math> at <math>P</math> and <math>Q</math>, respectively. Also <math>AB=5</math>, <math>BC=4</math>, <math>BQ=4.5</math>, and <math>BP=\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 +
 
 +
==Diagram==
 +
<asy>
 +
unitsize(30);
 +
draw(Circle((0,0),3));
 +
pair A,B,C,O, Q, P, M, N;
 +
A=(2.5, -sqrt(11/4));
 +
B=(-2.5, -sqrt(11/4));
 +
C=(-1.96, 2.28);
 +
Q=(-1.89, 2.81);
 +
P=(1.13, -1.68);
 +
O=origin;
 +
M=foot(O,C,B);
 +
N=foot(O,A,B);
 +
draw(A--B--C--cycle);
 +
label("$A$",A,SE);
 +
label("$B$",B,SW);
 +
label("$C$",C,NW);
 +
label("$Q$",Q,NW);
 +
dot(O);
 +
label("$O$",O,NE);
 +
label("$M$",M,W);
 +
label("$N$",N,S);
 +
label("$P$",P,S);
 +
draw(B--O);
 +
draw(C--Q);
 +
draw(Q--O);
 +
draw(O--C);
 +
draw(O--A);
 +
draw(O--P);
 +
draw(O--M, dashed);
 +
draw(O--N, dashed);
 +
draw(rightanglemark((-2.5, -sqrt(11/4)),(0,0),(-1.89, 2.81),5));
 +
draw(rightanglemark(O,N,B,5));
 +
draw(rightanglemark(B,O,P,5));
 +
draw(rightanglemark(O,M,C,5));
 +
</asy>
  
 
==Solution==
 
==Solution==
 +
 +
 +
===Solution 1===
 +
Call <math>M</math> and <math>N</math> the feet of the altitudes from <math>O</math> to <math>BC</math> and <math>AB</math>, respectively. Let <math>OB = r</math> . Notice that <math>\triangle{OMB} \sim \triangle{QOB}</math> because both are right triangles, and <math>\angle{OBQ} \cong \angle{OBM}</math>. By <math>\frac{MB}{BO}=\frac{BO}{BQ}</math>, <math>MB = r\left(\frac{r}{4.5}\right) = \frac{r^2}{4.5}</math>. However, since <math>O</math> is the circumcenter of triangle <math>ABC</math>, <math>OM</math> is a perpendicular bisector by the definition of a circumcenter. Hence, <math>\frac{r^2}{4.5} = 2 \implies r = 3</math>. Since we know <math>BN=\frac{5}{2}</math> and <math>\triangle BOP \sim \triangle BNO</math>, we have <math>\frac{BP}{3} = \frac{3}{\frac{5}{2}}</math>. Thus, <math>BP = \frac{18}{5}</math>.  <math>m + n=\boxed{023}</math>.
 +
 +
===Solution 2 (fastest)===
 +
Minor arc <math>BC = 2A</math> so <math>\angle{BOC}=2A</math>. Since <math>\triangle{BOC}</math> is isosceles (<math>BO</math> and <math>OC</math> are radii), <math>\angle{CBO}=(180-2A)/2=90-A</math>. <math>\angle{CBO}=90-A</math>, so <math>\angle{BQO}=A</math>. From this we get that <math>\triangle{BPQ}\sim \triangle{BCA}</math>. So <math>\dfrac{BP}{BC}=\dfrac{BQ}{BA}</math>, plugging in the given values we get <math>\dfrac{BP}{4}=\dfrac{4.5}{5}</math>, so <math>BP=\dfrac{18}{5}</math>, and <math>m+n=\boxed{023}</math>.
 +
 +
===Solution 3===
 +
Let <math>r=BO</math>. Drawing perpendiculars, <math>BM=MC=2</math> and <math>BN=NA=2.5</math>. From there, <cmath>OM=\sqrt{r^2-4}</cmath> Thus, <cmath>OQ=\frac{\sqrt{4r^2+9}}{2}</cmath> Using <math>\triangle{BOQ}</math>, we get <math>r=3</math>. Now let's find <math>NP</math>. After some calculations with <math>\triangle{BON}</math> ~ <math>\triangle{OPN}</math>, <math>{NP=11/10}</math>. Therefore, <cmath>BP=\frac{5}{2}+\frac{11}{10}=18/5</cmath> <math>18+5=\boxed{023}</math>.
 +
 +
===Solution 4===
 +
Let <math>\angle{BQO}=\alpha</math>. Extend <math>OB</math> to touch the circumcircle at a point <math>K</math>. Then, note that <math>\angle{KAC}=\angle{CBK}=\angle{QBO}=90^\circ-\alpha</math>. But since <math>BK</math> is a diameter, <math>\angle{KAB}=90^\circ</math>, implying <math>\angle{CAB}=\alpha</math>. It follows that <math>APCQ</math> is a cyclic quadrilateral.
 +
 +
Let <math>BP=x</math>. By Power of a Point, <cmath>5x=4\cdot\frac 9 2\implies x=\frac{18}{5}.</cmath>The answer is <math>18+5=\boxed{023}</math>.
 +
 +
===Solution 5===
 +
<math>\textit{Note: This is not a very good solution, but it is relatively natural and requires next to no thinking.}</math>
 +
 +
Denote the circumradius of <math>ABC</math> to be <math>R</math>, the circumcircle of <math>ABC</math> to be <math>O</math>, and the shortest distance from <math>Q</math> to circle <math>O</math> to be <math>x</math>. 
 +
 +
Using Power of a Point on <math>Q</math> relative to circle <math>O</math>, we get that <math>x(x+2r) = 0.5 \cdot 4.5 = \frac{9}{4}</math>. Using Pythagorean Theorem on triangle <math>QOB</math> to get <math>(x + r)^2 + r^2 = \frac{81}{4}</math>. Subtracting the first equation from the second, we get that <math>2r^2 = 18</math> and therefore <math>r = 3</math>. Now, set <math>\cos{ABC} = y</math>. Using law of cosines on <math>ABC</math> to find <math>AC</math> in terms of <math>y</math> and plugging that into the extended law of sines, we get <math>\frac{\sqrt{4^2 + 5^2 - 2 \cdot 4 \cdot 5 x}}{\sqrt{1 - x^2}} = 2R = 6</math>. Squaring both sides and cross multiplying, we get <math>36x^2 - 40x + 5 = 0</math>. Now, we get <math>x = \frac{10 \pm \sqrt{55}}{18}</math> using quadratic formula. If you drew a decent diagram, <math>B</math> is acute and therefore <math>x = \frac{10 + \sqrt{55}}{18}</math>(You can also try plugging in both in the end and seeing which gives a rational solution). Note that <math>BP = 3\frac{1}{\sin{OPB}} = \frac{3}{\cos{\angle ABC - \angle QBO}}.</math> Using the cosine addition formula and then plugging in what we know about <math>QBO</math>, we get that <math>BP = \frac{162}{2\cos{B} + \sqrt{5}\sin{B}}</math>. Now, the hard part is to find what <math>\sin{B}</math> is. We therefore want <math>\frac{\sqrt{324 - (10 + \sqrt{55})^2}}{18} = \frac{\sqrt{169 - 20\sqrt{55}}}{18}</math>. For the numerator, by inspection <math>(a + b\sqrt{55})^2</math> will not work for integers <math>a</math> and <math>b</math>. The other case is if there is <math>(a\sqrt{5} + b\sqrt{11})^2</math>. By inspection, <math>5\sqrt{5} - 2\sqrt{11}</math> works. Therefore, plugging all this in yields the answer, <math>\frac{18}{5} \rightarrow \boxed{23}</math>. Solution by hyxue
 +
 +
===Solution 6===
 +
<asy>
 +
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
 +
import graph; size(15cm);
 +
real labelscalefactor = 0.5; /* changes label-to-point distance */
 +
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
 +
pen dotstyle = black; /* point style */
 +
real xmin = -4.7673964645097335, xmax = 9.475267639476614, ymin = -1.6884766592324019, ymax = 6.385449160754665;  /* image dimensions */
 +
pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451);
 +
/* draw figures */
 +
draw(circle((0.7129306199257198,2.4781596958650733), 3.000319171815248), linewidth(2) + wrwrwr);
 +
draw((0.7129306199257198,2.4781596958650733)--(3.178984115621537,0.7692140299269852), linewidth(2) + wrwrwr);
 +
draw((xmin, 1.4430262733614363*xmin + 1.4493820802284032)--(xmax, 1.4430262733614363*xmax + 1.4493820802284032), linewidth(2) + wrwrwr); /* line */
 +
draw((xmin, -0.020161290322580634*xmin + 0.8333064516129032)--(xmax, -0.020161290322580634*xmax + 0.8333064516129032), linewidth(2) + wrwrwr); /* line */
 +
draw((xmin, -8.047527437688247*xmin + 26.352175924366414)--(xmax, -8.047527437688247*xmax + 26.352175924366414), linewidth(2) + wrwrwr); /* line */
 +
draw((xmin, -2.5113572383524088*xmin + 8.752778799300463)--(xmax, -2.5113572383524088*xmax + 8.752778799300463), linewidth(2) + wrwrwr); /* line */
 +
draw((xmin, 0.12426176956126818*xmin + 2.389569675458691)--(xmax, 0.12426176956126818*xmax + 2.389569675458691), linewidth(2) + wrwrwr); /* line */
 +
draw(circle((1.9173376033752174,4.895608471162773), 0.7842529827808445), linewidth(2) + wrwrwr);
 +
/* dots and labels */
 +
dot((-1.82,0.87),dotstyle);
 +
label("$A$", (-1.7801363959463627,0.965838014692327), NE * labelscalefactor);
 +
dot((3.178984115621537,0.7692140299269852),dotstyle);
 +
label("$B$", (3.2140445236332655,0.8641046996638531), NE * labelscalefactor);
 +
dot((2.6857306099246263,4.738685150758791),dotstyle);
 +
label("$C$", (2.7238749148597092,4.831703985774336), NE * labelscalefactor);
 +
dot((0.7129306199257198,2.4781596958650733),linewidth(4pt) + dotstyle);
 +
label("$O$", (0.7539479965810783,2.556577122410283), NE * labelscalefactor);
 +
dot((-0.42105034508654754,0.8417953698606159),linewidth(4pt) + dotstyle);
 +
label("$P$", (-0.38361543510094825,0.9195955987702934), NE * labelscalefactor);
 +
dot((2.6239558409689123,5.235819298886746),linewidth(4pt) + dotstyle);
 +
label("$Q$", (2.6591355325688624,5.312625111363486), NE * labelscalefactor);
 +
dot((1.3292769824200672,5.414489427724579),linewidth(4pt) + dotstyle);
 +
label("$A'$", (1.3643478867519216,5.488346291867214), NE * labelscalefactor);
 +
dot((1.8469115849379867,4.11452402186953),linewidth(4pt) + dotstyle);
 +
label("$P'$", (1.8822629450786978,4.184310162865866), NE * labelscalefactor);
 +
dot((2.5624172335003985,5.731052930966743),linewidth(4pt) + dotstyle);
 +
label("$D$", (2.603644633462422,5.802794720137042), NE * labelscalefactor);
 +
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
 +
</asy>
 +
Reflect <math>A</math>, <math>P</math> across <math>OB</math> to points <math>A'</math> and <math>P'</math>, respectively with <math>A'</math> on the circle and <math>P, O, P'</math> collinear. Now, <math>\angle A'CQ = 180^{\circ} - \angle A'CB = \angle A'AB = \angle P'PB</math> by parallel lines. From here, <math>\angle P'PB = \angle PP'B = \angle A'P'Q</math> as <math>P, P', Q</math> collinear. From here, <math>A'P'QC</math> is cyclic, and by power of a point we obtain <math>\frac{18}{5} \implies \boxed{023}</math>.
 +
~awang11's sol
 +
 +
==See also==
 +
{{AIME box|year=2015|n=II|num-b=10|num-a=12}}
 +
{{MAA Notice}}

Latest revision as of 20:59, 26 November 2023

Problem

The circumcircle of acute $\triangle ABC$ has center $O$. The line passing through point $O$ perpendicular to $\overline{OB}$ intersects lines $AB$ and $BC$ at $P$ and $Q$, respectively. Also $AB=5$, $BC=4$, $BQ=4.5$, and $BP=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Diagram

[asy] unitsize(30); draw(Circle((0,0),3)); pair A,B,C,O, Q, P, M, N; A=(2.5, -sqrt(11/4)); B=(-2.5, -sqrt(11/4)); C=(-1.96, 2.28); Q=(-1.89, 2.81); P=(1.13, -1.68); O=origin; M=foot(O,C,B); N=foot(O,A,B); draw(A--B--C--cycle); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$Q$",Q,NW); dot(O); label("$O$",O,NE); label("$M$",M,W); label("$N$",N,S); label("$P$",P,S); draw(B--O); draw(C--Q); draw(Q--O); draw(O--C); draw(O--A); draw(O--P); draw(O--M, dashed); draw(O--N, dashed); draw(rightanglemark((-2.5, -sqrt(11/4)),(0,0),(-1.89, 2.81),5)); draw(rightanglemark(O,N,B,5)); draw(rightanglemark(B,O,P,5)); draw(rightanglemark(O,M,C,5)); [/asy]

Solution

Solution 1

Call $M$ and $N$ the feet of the altitudes from $O$ to $BC$ and $AB$, respectively. Let $OB = r$ . Notice that $\triangle{OMB} \sim \triangle{QOB}$ because both are right triangles, and $\angle{OBQ} \cong \angle{OBM}$. By $\frac{MB}{BO}=\frac{BO}{BQ}$, $MB = r\left(\frac{r}{4.5}\right) = \frac{r^2}{4.5}$. However, since $O$ is the circumcenter of triangle $ABC$, $OM$ is a perpendicular bisector by the definition of a circumcenter. Hence, $\frac{r^2}{4.5} = 2 \implies r = 3$. Since we know $BN=\frac{5}{2}$ and $\triangle BOP \sim \triangle BNO$, we have $\frac{BP}{3} = \frac{3}{\frac{5}{2}}$. Thus, $BP = \frac{18}{5}$. $m + n=\boxed{023}$.

Solution 2 (fastest)

Minor arc $BC = 2A$ so $\angle{BOC}=2A$. Since $\triangle{BOC}$ is isosceles ($BO$ and $OC$ are radii), $\angle{CBO}=(180-2A)/2=90-A$. $\angle{CBO}=90-A$, so $\angle{BQO}=A$. From this we get that $\triangle{BPQ}\sim \triangle{BCA}$. So $\dfrac{BP}{BC}=\dfrac{BQ}{BA}$, plugging in the given values we get $\dfrac{BP}{4}=\dfrac{4.5}{5}$, so $BP=\dfrac{18}{5}$, and $m+n=\boxed{023}$.

Solution 3

Let $r=BO$. Drawing perpendiculars, $BM=MC=2$ and $BN=NA=2.5$. From there, \[OM=\sqrt{r^2-4}\] Thus, \[OQ=\frac{\sqrt{4r^2+9}}{2}\] Using $\triangle{BOQ}$, we get $r=3$. Now let's find $NP$. After some calculations with $\triangle{BON}$ ~ $\triangle{OPN}$, ${NP=11/10}$. Therefore, \[BP=\frac{5}{2}+\frac{11}{10}=18/5\] $18+5=\boxed{023}$.

Solution 4

Let $\angle{BQO}=\alpha$. Extend $OB$ to touch the circumcircle at a point $K$. Then, note that $\angle{KAC}=\angle{CBK}=\angle{QBO}=90^\circ-\alpha$. But since $BK$ is a diameter, $\angle{KAB}=90^\circ$, implying $\angle{CAB}=\alpha$. It follows that $APCQ$ is a cyclic quadrilateral.

Let $BP=x$. By Power of a Point, \[5x=4\cdot\frac 9 2\implies x=\frac{18}{5}.\]The answer is $18+5=\boxed{023}$.

Solution 5

$\textit{Note: This is not a very good solution, but it is relatively natural and requires next to no thinking.}$

Denote the circumradius of $ABC$ to be $R$, the circumcircle of $ABC$ to be $O$, and the shortest distance from $Q$ to circle $O$ to be $x$.

Using Power of a Point on $Q$ relative to circle $O$, we get that $x(x+2r) = 0.5 \cdot 4.5 = \frac{9}{4}$. Using Pythagorean Theorem on triangle $QOB$ to get $(x + r)^2 + r^2 = \frac{81}{4}$. Subtracting the first equation from the second, we get that $2r^2 = 18$ and therefore $r = 3$. Now, set $\cos{ABC} = y$. Using law of cosines on $ABC$ to find $AC$ in terms of $y$ and plugging that into the extended law of sines, we get $\frac{\sqrt{4^2 + 5^2 - 2 \cdot 4 \cdot 5 x}}{\sqrt{1 - x^2}} = 2R = 6$. Squaring both sides and cross multiplying, we get $36x^2 - 40x + 5 = 0$. Now, we get $x = \frac{10 \pm \sqrt{55}}{18}$ using quadratic formula. If you drew a decent diagram, $B$ is acute and therefore $x = \frac{10 + \sqrt{55}}{18}$(You can also try plugging in both in the end and seeing which gives a rational solution). Note that $BP = 3\frac{1}{\sin{OPB}} = \frac{3}{\cos{\angle ABC - \angle QBO}}.$ Using the cosine addition formula and then plugging in what we know about $QBO$, we get that $BP = \frac{162}{2\cos{B} + \sqrt{5}\sin{B}}$. Now, the hard part is to find what $\sin{B}$ is. We therefore want $\frac{\sqrt{324 - (10 + \sqrt{55})^2}}{18} = \frac{\sqrt{169 - 20\sqrt{55}}}{18}$. For the numerator, by inspection $(a + b\sqrt{55})^2$ will not work for integers $a$ and $b$. The other case is if there is $(a\sqrt{5} + b\sqrt{11})^2$. By inspection, $5\sqrt{5} - 2\sqrt{11}$ works. Therefore, plugging all this in yields the answer, $\frac{18}{5} \rightarrow \boxed{23}$. Solution by hyxue

Solution 6

[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(15cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.7673964645097335, xmax = 9.475267639476614, ymin = -1.6884766592324019, ymax = 6.385449160754665; /* image dimensions */ pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); /* draw figures */ draw(circle((0.7129306199257198,2.4781596958650733), 3.000319171815248), linewidth(2) + wrwrwr); draw((0.7129306199257198,2.4781596958650733)--(3.178984115621537,0.7692140299269852), linewidth(2) + wrwrwr); draw((xmin, 1.4430262733614363*xmin + 1.4493820802284032)--(xmax, 1.4430262733614363*xmax + 1.4493820802284032), linewidth(2) + wrwrwr); /* line */ draw((xmin, -0.020161290322580634*xmin + 0.8333064516129032)--(xmax, -0.020161290322580634*xmax + 0.8333064516129032), linewidth(2) + wrwrwr); /* line */ draw((xmin, -8.047527437688247*xmin + 26.352175924366414)--(xmax, -8.047527437688247*xmax + 26.352175924366414), linewidth(2) + wrwrwr); /* line */ draw((xmin, -2.5113572383524088*xmin + 8.752778799300463)--(xmax, -2.5113572383524088*xmax + 8.752778799300463), linewidth(2) + wrwrwr); /* line */ draw((xmin, 0.12426176956126818*xmin + 2.389569675458691)--(xmax, 0.12426176956126818*xmax + 2.389569675458691), linewidth(2) + wrwrwr); /* line */ draw(circle((1.9173376033752174,4.895608471162773), 0.7842529827808445), linewidth(2) + wrwrwr); /* dots and labels */ dot((-1.82,0.87),dotstyle); label("$A$", (-1.7801363959463627,0.965838014692327), NE * labelscalefactor); dot((3.178984115621537,0.7692140299269852),dotstyle); label("$B$", (3.2140445236332655,0.8641046996638531), NE * labelscalefactor); dot((2.6857306099246263,4.738685150758791),dotstyle); label("$C$", (2.7238749148597092,4.831703985774336), NE * labelscalefactor); dot((0.7129306199257198,2.4781596958650733),linewidth(4pt) + dotstyle); label("$O$", (0.7539479965810783,2.556577122410283), NE * labelscalefactor); dot((-0.42105034508654754,0.8417953698606159),linewidth(4pt) + dotstyle); label("$P$", (-0.38361543510094825,0.9195955987702934), NE * labelscalefactor); dot((2.6239558409689123,5.235819298886746),linewidth(4pt) + dotstyle); label("$Q$", (2.6591355325688624,5.312625111363486), NE * labelscalefactor); dot((1.3292769824200672,5.414489427724579),linewidth(4pt) + dotstyle); label("$A'$", (1.3643478867519216,5.488346291867214), NE * labelscalefactor); dot((1.8469115849379867,4.11452402186953),linewidth(4pt) + dotstyle); label("$P'$", (1.8822629450786978,4.184310162865866), NE * labelscalefactor); dot((2.5624172335003985,5.731052930966743),linewidth(4pt) + dotstyle); label("$D$", (2.603644633462422,5.802794720137042), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] Reflect $A$, $P$ across $OB$ to points $A'$ and $P'$, respectively with $A'$ on the circle and $P, O, P'$ collinear. Now, $\angle A'CQ = 180^{\circ} - \angle A'CB = \angle A'AB = \angle P'PB$ by parallel lines. From here, $\angle P'PB = \angle PP'B = \angle A'P'Q$ as $P, P', Q$ collinear. From here, $A'P'QC$ is cyclic, and by power of a point we obtain $\frac{18}{5} \implies \boxed{023}$. ~awang11's sol

See also

2015 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2015_AIME_II_Problems/Problem_11&oldid=205981"