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> | + | 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== | ==Diagram== | ||
Line 39: | Line 39: | ||
draw(rightanglemark(O,M,C,5)); | draw(rightanglemark(O,M,C,5)); | ||
</asy> | </asy> | ||
+ | |||
+ | ==Solution== | ||
+ | |||
+ | |||
===Solution 1=== | ===Solution 1=== | ||
− | Call | + | 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 2=== | + | ===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== | ==See also== | ||
{{AIME box|year=2015|n=II|num-b=10|num-a=12}} | {{AIME box|year=2015|n=II|num-b=10|num-a=12}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 21:59, 26 November 2023
Contents
Problem
The circumcircle of acute has center
. The line passing through point
perpendicular to
intersects lines
and
at
and
, respectively. Also
,
,
, and
, where
and
are relatively prime positive integers. Find
.
Diagram
Solution
Solution 1
Call and
the feet of the altitudes from
to
and
, respectively. Let
. Notice that
because both are right triangles, and
. By
,
. However, since
is the circumcenter of triangle
,
is a perpendicular bisector by the definition of a circumcenter. Hence,
. Since we know
and
, we have
. Thus,
.
.
Solution 2 (fastest)
Minor arc so
. Since
is isosceles (
and
are radii),
.
, so
. From this we get that
. So
, plugging in the given values we get
, so
, and
.
Solution 3
Let . Drawing perpendiculars,
and
. From there,
Thus,
Using
, we get
. Now let's find
. After some calculations with
~
,
. Therefore,
.
Solution 4
Let . Extend
to touch the circumcircle at a point
. Then, note that
. But since
is a diameter,
, implying
. It follows that
is a cyclic quadrilateral.
Let . By Power of a Point,
The answer is
.
Solution 5
Denote the circumradius of to be
, the circumcircle of
to be
, and the shortest distance from
to circle
to be
.
Using Power of a Point on relative to circle
, we get that
. Using Pythagorean Theorem on triangle
to get
. Subtracting the first equation from the second, we get that
and therefore
. Now, set
. Using law of cosines on
to find
in terms of
and plugging that into the extended law of sines, we get
. Squaring both sides and cross multiplying, we get
. Now, we get
using quadratic formula. If you drew a decent diagram,
is acute and therefore
(You can also try plugging in both in the end and seeing which gives a rational solution). Note that
Using the cosine addition formula and then plugging in what we know about
, we get that
. Now, the hard part is to find what
is. We therefore want
. For the numerator, by inspection
will not work for integers
and
. The other case is if there is
. By inspection,
works. Therefore, plugging all this in yields the answer,
. Solution by hyxue
Solution 6
Reflect
,
across
to points
and
, respectively with
on the circle and
collinear. Now,
by parallel lines. From here,
as
collinear. From here,
is cyclic, and by power of a point we obtain
.
~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.