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Difference between revisions of "2009 AIME I Problems/Problem 4"

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== Problem 4 ==
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== Problem==
 
In parallelogram <math>ABCD</math>, point <math>M</math> is on <math>\overline{AB}</math> so that <math>\frac {AM}{AB} = \frac {17}{1000}</math> and point <math>N</math> is on <math>\overline{AD}</math> so that <math>\frac {AN}{AD} = \frac {17}{2009}</math>. Let <math>P</math> be the point of intersection of <math>\overline{AC}</math> and <math>\overline{MN}</math>. Find <math>\frac {AC}{AP}</math>.
 
In parallelogram <math>ABCD</math>, point <math>M</math> is on <math>\overline{AB}</math> so that <math>\frac {AM}{AB} = \frac {17}{1000}</math> and point <math>N</math> is on <math>\overline{AD}</math> so that <math>\frac {AN}{AD} = \frac {17}{2009}</math>. Let <math>P</math> be the point of intersection of <math>\overline{AC}</math> and <math>\overline{MN}</math>. Find <math>\frac {AC}{AP}</math>.
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[[2009 AIME I Problems/Problem 4|Solution]]
 
  
 
==Solution 1==
 
==Solution 1==
  
 
One of the ways to solve this problem is to make this parallelogram a straight line.
 
One of the ways to solve this problem is to make this parallelogram a straight line.
 +
So the whole length of the line is <math>APC</math>(<math>AMC</math> or <math>ANC</math>), and <math>ABC</math> is <math>1000x+2009x=3009x.</math>
  
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So the whole length of the line <math>APC</math>(<math>AMC</math> or <math>ANC</math>), and <math>ABC</math> is <math>1000x+2009x=3009x</math>
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<math>AP</math>(<math>AM</math> or <math>AN</math>) is <math>17x.</math>
  
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And <math>AP</math>(<math>AM</math> or <math>AN</math>) is <math>17x</math>
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So the answer is <math>3009x/17x = \boxed{177}</math>
  
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So the answer is <math>3009x/17x = \boxed{177}</math>
 
 
==Solution 2==
 
==Solution 2==
 +
 
Draw a diagram with all the given points and lines involved. Construct parallel lines <math>\overline{DF_2F_1}</math> and <math>\overline{BB_1B_2}</math> to <math>\overline{MN}</math>, where for the lines the endpoints are on <math>\overline{AM}</math> and <math>\overline{AN}</math>, respectively, and each point refers to an intersection. Also, draw the median of quadrilateral <math>BB_2DF_1</math> <math>\overline{E_1E_2E_3}</math> where the points are in order from top to bottom. Clearly, by similar triangles, <math>BB_2 = \frac {1000}{17}MN</math> and <math>DF_1 = \frac {2009}{17}MN</math>. It is not difficult to see that <math>E_2</math> is the center of quadrilateral <math>ABCD</math> and thus the midpoint of <math>\overline{AC}</math> as well as the midpoint of <math>\overline{B_1}{F_2}</math> (all of this is easily proven with symmetry). From more triangle similarity, <math>E_1E_3 = \frac12\cdot\frac {3009}{17}MN\implies AE_2 = \frac12\cdot\frac {3009}{17}AP\implies AC = 2\cdot\frac12\cdot\frac {3009}{17}AP</math>
 
Draw a diagram with all the given points and lines involved. Construct parallel lines <math>\overline{DF_2F_1}</math> and <math>\overline{BB_1B_2}</math> to <math>\overline{MN}</math>, where for the lines the endpoints are on <math>\overline{AM}</math> and <math>\overline{AN}</math>, respectively, and each point refers to an intersection. Also, draw the median of quadrilateral <math>BB_2DF_1</math> <math>\overline{E_1E_2E_3}</math> where the points are in order from top to bottom. Clearly, by similar triangles, <math>BB_2 = \frac {1000}{17}MN</math> and <math>DF_1 = \frac {2009}{17}MN</math>. It is not difficult to see that <math>E_2</math> is the center of quadrilateral <math>ABCD</math> and thus the midpoint of <math>\overline{AC}</math> as well as the midpoint of <math>\overline{B_1}{F_2}</math> (all of this is easily proven with symmetry). From more triangle similarity, <math>E_1E_3 = \frac12\cdot\frac {3009}{17}MN\implies AE_2 = \frac12\cdot\frac {3009}{17}AP\implies AC = 2\cdot\frac12\cdot\frac {3009}{17}AP</math>
 
<math>= \boxed{177}AP</math>.
 
<math>= \boxed{177}AP</math>.
  
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A diagram would be appreciated (and whoever makes it can take out the descriptive stuff).
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==Solution 3==
 +
 
 +
Using vectors, note that <math>\overrightarrow{AM}=\frac{17}{1000}\overrightarrow{AB}</math> and <math>\overrightarrow{AN}=\frac{17}{2009}\overrightarrow{AD}</math>.  Note that <math>\overrightarrow{AP}=\frac{x\overrightarrow{AM}+y\overrightarrow{AN}}{x+y}</math> for some positive x and y, but at the same time is a scalar multiple of <math>\overrightarrow{AB}+\overrightarrow{AD}</math>.  So, writing the equation <math>\overrightarrow{AP}=\frac{x\overrightarrow{AM}+y\overrightarrow{AN}}{x+y}</math> in terms of <math>\overrightarrow{AB}</math> and <math>\overrightarrow{AD}</math>, we have <math>\overrightarrow{AP}=\frac{\frac{17x}{1000}\overrightarrow{AB}+\frac{17y}{2009}\overrightarrow{AD}}{x+y}</math>.  But the coefficients of the two vectors must be equal because, as already stated, <math>\overrightarrow{AP}</math> is a scalar multiple of <math>\overrightarrow{AB}+\overrightarrow{AD}</math>.  We then see that <math>\frac{x}{x+y}=\frac{1000}{3009}</math> and <math>\frac{y}{x+y}=\frac{2009}{3009}</math>. Finally, we have <math>\overrightarrow{AP}=\frac{17}{3009}(\overrightarrow{AB}+\overrightarrow{AD})</math> and, simplifying, <math>\overrightarrow{AB}+\overrightarrow{AD}=177\overrightarrow{AP}</math> and the desired quantity is <math>177</math>.
 +
 
 +
==Solution 4==
 +
 
 +
We approach the problem using mass points on triangle <math>ABD</math> as displayed below.
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<asy>
 +
 
 +
pair A=(0,0),B=(50,0),D=(10,40),C=B+D,M=(8,0),NN=(2,8);
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draw(A--B--C--D--cycle);
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draw(B--D^^A--C^^M--NN);
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pair O=extension(A,C,B,D);
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pair P=extension(A,C,M,NN);
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dot(A);dot(B);dot(C);dot(D);dot(O);dot(M);dot(NN);dot(P);
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label("$A$",A,SW);
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label("$B$",B,SE);
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label("$C$",C,NE);
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label("$D$",D,NW);
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label("$M$",M,S);
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label("$N$",NN,NW);
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label("$P$",P,NNE);
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label("$O$",O,N);
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</asy>
 +
 
 +
But as <math>MN</math> does not protrude from a vertex, we will have to "split the mass" at point <math>A</math>. First, we know that <math>DO</math>  is congruent to <math>BO</math> because diagonals of parallelograms bisect each other. Therefore, we can assign equal masses to points <math>B</math> and <math>D</math>. In this case, we assign <math>B</math> and <math>D</math> a mass of 17 each. Now we split the mass at <math>A</math>, so we balance segments <math>AB</math> and <math>AD</math> separately, and then the mass of <math>A</math> is the sum of those masses. A mass of 983 is required to balance segment <math>AB</math>, while a mass of 1992 is required to balance segment <math>AD</math>. Therefore, <math>A</math> has a mass of <math>1992+983=2975</math>. Also, <math>O</math> has a mass of 34.  Therefore, <math>\frac{AO}{AP}=\frac{2975+34}{34}=\frac{3009}{34}</math>, so <math>\frac{AC}{AP}=\frac{2 (3009)}{34}=177</math>.
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 +
==Solution 5==
 +
 
 +
Assume, for the ease of computation, that <math>AM=AN=17</math>, <math>AB=1000</math>, and <math>AD=2009</math>.  Now, let line <math>MN</math> intersect line <math>CD</math> at point <math>X</math> and let <math>Y</math> be a point such that <math>XY\parallel AD</math> and <math>AY\parallel DX</math>.  As a result, <math>ADXY</math> is a parallelogram.  By construction, <math>\triangle MAN\sim \triangle MYX</math> so <cmath>\frac{MY}{MA}=\frac{YX}{AN}=\frac{AD}{AN}=\frac{2009}{17}\implies MY=2009</cmath> and <math>AY=DX=2009-17</math>.  Also, because <math>AM\parallel XC</math>, we have <math>\triangle PAM\sim \triangle PCX</math> so <cmath>\frac{PC}{PA}=\frac{CX}{AM}=\frac{DX+CD}{AM}=\frac{2009-17+1000}{17}=176.</cmath>  Hence, <math>\frac{AC}{AP}=\frac{PC}{PA}+1=\boxed{177}.</math>
 +
 
 +
==Solution 6(Coordinate Geometry)==
 +
 
 +
Assign <math>A = (0,0)</math>. Since there are no constraints in the problem against this, assume <math>ABCD</math> to be a rectangle with dimensions <math>1000 \times 2009.</math> Now, we can assign <cmath>A=(0,0)</cmath> <cmath>B=(1000, 0)</cmath> <cmath>C=(1000,-2009)</cmath> <cmath>D=(0,-2009).</cmath>
 +
 
 +
Then, since <math>\frac{AM}{AB} = \frac{17}{1000}</math> and <math>AB = 1000</math>, we can place <math>M</math> at <math>(17, 0).</math> Similarly, place <math>AN</math> at <math>(0, 17).</math>
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Then, the equation of line <math>MN</math> is <math>y=x-17,</math> and the equation of <math>AC</math> is <math>y=\frac{-2009}{1000}x.</math> Solve to find point <math>P</math> at <math>\left( \frac{1000}{177}, \frac{-2009}{177} \right)</math>.
 +
 
 +
We can calculate vectors to represent the distances: <cmath>\overrightarrow{AC}= <1000, -2009></cmath> <cmath>\overrightarrow{AP}= \frac{1}{177}<1000, -2009>.</cmath> In this way, we can see that <cmath>AC:AP = 177:1,</cmath> and our answer is <math>\boxed{177}.</math>
 +
 
 +
~ [https://artofproblemsolving.com/community/user/357326 HappyHuman]
 +
 
 +
===Note===
 +
It is possible to use [[coordinate geometry]] without setting <math>ABCD</math> as a rectangle, either by projecting the plane onto another (tilted) plane or removing the restriction that the [[axes]] have to be [[perpendicular]].
 +
 
 +
==Video Solution==
 +
Unique solution: https://youtu.be/2Xzjh6ae0MU
 +
 
 +
~IceMatrix
 +
 
 +
==Video Solution==
 +
https://youtu.be/kALrIDMR0dg
 +
 
 +
~Shreyas S
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2009|n=I|num-b=3|num-a=5}}
 
{{AIME box|year=2009|n=I|num-b=3|num-a=5}}
 +
[[Category:Intermediate Geometry Problems]]
 +
{{MAA Notice}}

Latest revision as of 23:23, 8 January 2023

Problem

In parallelogram $ABCD$, point $M$ is on $\overline{AB}$ so that $\frac {AM}{AB} = \frac {17}{1000}$ and point $N$ is on $\overline{AD}$ so that $\frac {AN}{AD} = \frac {17}{2009}$. Let $P$ be the point of intersection of $\overline{AC}$ and $\overline{MN}$. Find $\frac {AC}{AP}$.

Solution 1

One of the ways to solve this problem is to make this parallelogram a straight line. So the whole length of the line is $APC$($AMC$ or $ANC$), and $ABC$ is $1000x+2009x=3009x.$

$AP$($AM$ or $AN$) is $17x.$

So the answer is $3009x/17x = \boxed{177}$

Solution 2

Draw a diagram with all the given points and lines involved. Construct parallel lines $\overline{DF_2F_1}$ and $\overline{BB_1B_2}$ to $\overline{MN}$, where for the lines the endpoints are on $\overline{AM}$ and $\overline{AN}$, respectively, and each point refers to an intersection. Also, draw the median of quadrilateral $BB_2DF_1$ $\overline{E_1E_2E_3}$ where the points are in order from top to bottom. Clearly, by similar triangles, $BB_2 = \frac {1000}{17}MN$ and $DF_1 = \frac {2009}{17}MN$. It is not difficult to see that $E_2$ is the center of quadrilateral $ABCD$ and thus the midpoint of $\overline{AC}$ as well as the midpoint of $\overline{B_1}{F_2}$ (all of this is easily proven with symmetry). From more triangle similarity, $E_1E_3 = \frac12\cdot\frac {3009}{17}MN\implies AE_2 = \frac12\cdot\frac {3009}{17}AP\implies AC = 2\cdot\frac12\cdot\frac {3009}{17}AP$ $= \boxed{177}AP$.

Solution 3

Using vectors, note that $\overrightarrow{AM}=\frac{17}{1000}\overrightarrow{AB}$ and $\overrightarrow{AN}=\frac{17}{2009}\overrightarrow{AD}$. Note that $\overrightarrow{AP}=\frac{x\overrightarrow{AM}+y\overrightarrow{AN}}{x+y}$ for some positive x and y, but at the same time is a scalar multiple of $\overrightarrow{AB}+\overrightarrow{AD}$. So, writing the equation $\overrightarrow{AP}=\frac{x\overrightarrow{AM}+y\overrightarrow{AN}}{x+y}$ in terms of $\overrightarrow{AB}$ and $\overrightarrow{AD}$, we have $\overrightarrow{AP}=\frac{\frac{17x}{1000}\overrightarrow{AB}+\frac{17y}{2009}\overrightarrow{AD}}{x+y}$. But the coefficients of the two vectors must be equal because, as already stated, $\overrightarrow{AP}$ is a scalar multiple of $\overrightarrow{AB}+\overrightarrow{AD}$. We then see that $\frac{x}{x+y}=\frac{1000}{3009}$ and $\frac{y}{x+y}=\frac{2009}{3009}$. Finally, we have $\overrightarrow{AP}=\frac{17}{3009}(\overrightarrow{AB}+\overrightarrow{AD})$ and, simplifying, $\overrightarrow{AB}+\overrightarrow{AD}=177\overrightarrow{AP}$ and the desired quantity is $177$.

Solution 4

We approach the problem using mass points on triangle $ABD$ as displayed below. [asy] pair A=(0,0),B=(50,0),D=(10,40),C=B+D,M=(8,0),NN=(2,8); draw(A--B--C--D--cycle); draw(B--D^^A--C^^M--NN); pair O=extension(A,C,B,D); pair P=extension(A,C,M,NN); dot(A);dot(B);dot(C);dot(D);dot(O);dot(M);dot(NN);dot(P); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$M$",M,S); label("$N$",NN,NW); label("$P$",P,NNE); label("$O$",O,N); [/asy]

But as $MN$ does not protrude from a vertex, we will have to "split the mass" at point $A$. First, we know that $DO$ is congruent to $BO$ because diagonals of parallelograms bisect each other. Therefore, we can assign equal masses to points $B$ and $D$. In this case, we assign $B$ and $D$ a mass of 17 each. Now we split the mass at $A$, so we balance segments $AB$ and $AD$ separately, and then the mass of $A$ is the sum of those masses. A mass of 983 is required to balance segment $AB$, while a mass of 1992 is required to balance segment $AD$. Therefore, $A$ has a mass of $1992+983=2975$. Also, $O$ has a mass of 34. Therefore, $\frac{AO}{AP}=\frac{2975+34}{34}=\frac{3009}{34}$, so $\frac{AC}{AP}=\frac{2 (3009)}{34}=177$.

Solution 5

Assume, for the ease of computation, that $AM=AN=17$, $AB=1000$, and $AD=2009$. Now, let line $MN$ intersect line $CD$ at point $X$ and let $Y$ be a point such that $XY\parallel AD$ and $AY\parallel DX$. As a result, $ADXY$ is a parallelogram. By construction, $\triangle MAN\sim \triangle MYX$ so \[\frac{MY}{MA}=\frac{YX}{AN}=\frac{AD}{AN}=\frac{2009}{17}\implies MY=2009\] and $AY=DX=2009-17$. Also, because $AM\parallel XC$, we have $\triangle PAM\sim \triangle PCX$ so \[\frac{PC}{PA}=\frac{CX}{AM}=\frac{DX+CD}{AM}=\frac{2009-17+1000}{17}=176.\] Hence, $\frac{AC}{AP}=\frac{PC}{PA}+1=\boxed{177}.$

Solution 6(Coordinate Geometry)

Assign $A = (0,0)$. Since there are no constraints in the problem against this, assume $ABCD$ to be a rectangle with dimensions $1000 \times 2009.$ Now, we can assign \[A=(0,0)\] \[B=(1000, 0)\] \[C=(1000,-2009)\] \[D=(0,-2009).\]

Then, since $\frac{AM}{AB} = \frac{17}{1000}$ and $AB = 1000$, we can place $M$ at $(17, 0).$ Similarly, place $AN$ at $(0, 17).$ Then, the equation of line $MN$ is $y=x-17,$ and the equation of $AC$ is $y=\frac{-2009}{1000}x.$ Solve to find point $P$ at $\left( \frac{1000}{177}, \frac{-2009}{177} \right)$.

We can calculate vectors to represent the distances: \[\overrightarrow{AC}= <1000, -2009>\] \[\overrightarrow{AP}= \frac{1}{177}<1000, -2009>.\] In this way, we can see that \[AC:AP = 177:1,\] and our answer is $\boxed{177}.$

~ HappyHuman

Note

It is possible to use coordinate geometry without setting $ABCD$ as a rectangle, either by projecting the plane onto another (tilted) plane or removing the restriction that the axes have to be perpendicular.

Video Solution

Unique solution: https://youtu.be/2Xzjh6ae0MU

~IceMatrix

Video Solution

https://youtu.be/kALrIDMR0dg

~Shreyas S

See also

2009 AIME I (Problems • Answer Key • Resources)
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