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Difference between revisions of "2024 AIME I Problems/Problem 8"
(→‎Video Solution（Chinese）subtitle in English)
 
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Eight circles of radius <math>34</math> are sequentially tangent, and two of the circles are tangent to <math>AB</math> and <math>BC</math> of triangle <math>ABC</math>, respectively. <math>2024</math> circles of radius <math>1</math> can be arranged in the same manner. The inradius of triangle <math>ABC</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 
Eight circles of radius <math>34</math> are sequentially tangent, and two of the circles are tangent to <math>AB</math> and <math>BC</math> of triangle <math>ABC</math>, respectively. <math>2024</math> circles of radius <math>1</math> can be arranged in the same manner. The inradius of triangle <math>ABC</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  
−
==Solution==
+
<asy>
−
Notice that the incircle is the same as a case with one circle of <math>x</math> radius.
+
pair A = (2,1); pair B = (0,0); pair C = (3,0);
 +
dot(A^^B^^C);
 +
label("$A$", A, N); label("$B$", B, S); label("$C$", C, S);
 +
draw(A--B--C--cycle);
 +
for(real i=0.62; i<2.7; i+=0.29){
 +
draw(circle((i,0.145), 0.145));
 +
}
 +
</asy>
 +
 
 +
==Solution 1==
 +
Draw an altitude from both end circles of the diagram with the circles of radius one, and call the lengths you get drawing the altitudes of the circles down to <math>BC</math> <math>a</math> and <math>b</math>. Now we have the length of side <math>BC</math> of being <math>(2)(2022)+1+1+a+b</math>. However, the side <math>BC</math> can also be written as <math>(6)(68)+34+34+34a+34b</math>, due to similar triangles from the second diagram. If we set the equations equal, we have <math>\frac{1190}{11} = a+b</math>. Call the radius of the incircle <math>r</math>, then we have the side BC to be <math>r(a+b)</math>. We find <math>r</math> as <math>\frac{4046+\frac{1190}{11}}{\frac{1190}{11}}</math>, which simplifies to <math>\frac{10+((34)(11))}{10}</math>,so we have <math>\frac{192}{5}</math>, which sums to <math>\boxed{197}</math>.
 +
 
 +
==Solution 2==
 +
Assume that <math>ABC</math> is isosceles with <math>AB=AC</math>.
 +
 
 +
If we let <math>P_1</math> be the intersection of <math>BC</math> and the leftmost of the eight circles of radius <math>34</math>, <math>N_1</math> the center of the leftmost circle, and <math>M_1</math> the intersection of the leftmost circle and <math>AB</math>, and we do the same for the <math>2024</math> circles of radius <math>1</math>, naming the points <math>P_2</math>, <math>N_2</math>, and <math>M_2</math>, respectively, then we see that <math>BP_1N_1M_1\sim BP_2N_2M_2</math>. The same goes for vertex <math>C</math>, and the corresponding quadrilaterals are congruent.
 +
 
 +
Let <math>x=BP_2</math>. We see that <math>BP_1=34x</math> by similarity ratios (due to the radii). The corresponding figures on vertex <math>C</math> are also these values. If we combine the distances of the figures, we see that <math>BC=2x+4046</math> and <math>BC=68x+476</math>, and solving this system, we find that <math>x=\frac{595}{11}</math>.
 +
 
 +
If we consider that the incircle of <math>\triangle ABC</math> is essentially the case of <math>1</math> circle with <math>r</math> radius (the inradius of <math>\triangle ABC</math>, we can find that <math>BC=2rx</math>. From <math>BC=2x+4046</math>, we have:
 +
 
 +
<math>r=1+\frac{2023}{x}</math>
 +
 
 +
<math>=1+\frac{11\cdot2023}{595}</math>
 +
 
 +
<math>=1+\frac{187}{5}</math>
 +
 
 +
<math>=\frac{192}{5}</math>
 +
 
 +
Thus the answer is <math>192+5=\boxed{197}</math>.
 +
 
 +
~eevee9406
 +
 
 +
==Solution 3==
 +
 
 +
Let <math>x = \cot{\frac{B}{2}} + \cot{\frac{C}{2}}</math>. By representing <math>BC</math> in two ways, we have the following:
 +
<cmath>34x + 7\cdot 34\cdot 2 = BC</cmath>
 +
<cmath>x + 2023 \cdot 2 = BC</cmath>
 +
 
 +
Solving we find <math>x = \frac{1190}{11}</math>.
 +
Now draw the inradius, let it be <math>r</math>. We find that <math>rx =BC</math>, hence
 +
<cmath>xr = x + 4046 \implies r-1 = \frac{11}{1190}\cdot 4046 = \frac{187}{5}.</cmath>
 +
Thus <cmath>r = \frac{192}{5} \implies \boxed{197}.</cmath>
 +
~AtharvNaphade
 +
 
 +
==Solution 4==
 +
 
 +
First, let the circle tangent to <math>AB</math> and <math>BC</math> be <math>O</math> and the other circle that is tangent to <math>AC</math> and <math>BC</math> be <math>R</math>. Let <math>x</math> be the distance from the tangency point on line segment <math>BC</math> of the circle <math>O</math> to <math>B</math>. Also, let <math>y</math> be the distance of the tangency point of circle <math>R</math> on the line segment <math>BC</math> to point <math>C</math>. Realize that we can let <math>n</math> be the number of circles tangent to line segment <math>BC</math> and <math>r</math> be the corresponding radius of each of the circles. Also, the circles that are tangent to <math>BC</math> are similar. So, we can build the equation <math>BC = (x+y+2(n-1)) \times r</math>. Looking at the given information, we see that when <math>n=8</math>, <math>r=34</math>, and when <math>n=2024</math>, <math>r=1</math>, and we also want to find the radius <math>r</math> in the case where <math>n=1</math>. Using these facts, we can write the following equations:
 +
 
 +
<math>BC = (x+y+2(8-1)) \times 34 = (x+y+2(2024-1)) \times 1 = (x+y+2(1-1)) \times r</math>
 +
 
 +
We can find that <math>x+y = \frac{1190}{11}</math> . Now, let <math>(x+y+2(2024-1)) \times 1 = (x+y+2(1-1)) \times r</math>.
 +
 
 +
Substituting <math>x+y = \frac{1190}{11}</math> in, we find that <cmath>r = \frac{192}{5} \implies \boxed{197}.</cmath>
 +
 
 +
~EaZ_Shadow
 +
 
 +
==Solution 5 (one variable)==
 +
 
 +
Define <math>I, x_1, x_8, y_1, y_{2024}</math> to be the incenter and centers of the first and last circles of the <math>8</math> and <math>2024</math> tangent circles to <math>BC,</math> and define <math>r</math> to be the inradius of triangle <math>\bigtriangleup ABC.</math> We calculate <math>\overline{x_1x_8} = 34 \cdot 14</math> and <math>\overline{y_1y_{2024}} = 1 \cdot 4046</math> because connecting the center of the circles voids two extra radii.
 +
 
 +
 
 +
We can easily see that <math>B, x_1, x_8,</math> and <math>I</math> are collinear, and the same follows for <math>C, y_1, y_2024,</math> and <math>I</math> (think angle bisectors).
 +
 
 +
 
 +
We observe that triangles <math>\bigtriangleup I x_1 x_8</math> and <math>\bigtriangleup I y_1 y_{2024}</math> are similar, and therefore the ratio of the altitude to the base is the same, so we note
 +
 
 +
<cmath>\frac{\text{altitude}}{\text{base}} = \frac{r-34}{34\cdot 14} = \frac{r-1}{1\cdot 4046}.</cmath>
 +
 
 +
 
 +
Solving yields <math>r = \frac{192}{5},</math> so the answer is <math>192+5 = \boxed{197}.</math>
 +
 
 +
-[https://artofproblemsolving.com/wiki/index.php/User:Spectraldragon8 spectraldragon8]
 +
 
 +
==Video Solution（Chinese）subtitle in English==
 +
https://youtu.be/q8N-zzlUFpA
 +
 
 +
==Video Solution 1 by OmegaLearn.org==
 +
https://youtu.be/MWTf6Jr8UwU
 +
 
 +
==Video Solution==
 +
 
 +
https://youtu.be/qlrVguS79GA
 +
 
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 +
 
 +
==Video Solution==
 +
https://www.youtube.com/watch?v=MWhLgPr-ZR8&t=716s&ab_channel=TheBeautyofMath (beautyofmath)
 +
 
 +
==See also==
 +
{{AIME box|year=2024|n=I|num-b=7|num-a=9}}
 +
 
 +
{{MAA Notice}}

Latest revision as of 03:32, 25 December 2024

Problem

Eight circles of radius $34$ are sequentially tangent, and two of the circles are tangent to $AB$ and $BC$ of triangle $ABC$, respectively. $2024$ circles of radius $1$ can be arranged in the same manner. The inradius of triangle $ABC$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

[asy] pair A = (2,1); pair B = (0,0); pair C = (3,0); dot(A^^B^^C); label("$A$", A, N); label("$B$", B, S); label("$C$", C, S); draw(A--B--C--cycle); for(real i=0.62; i<2.7; i+=0.29){ draw(circle((i,0.145), 0.145)); } [/asy]

Solution 1

Draw an altitude from both end circles of the diagram with the circles of radius one, and call the lengths you get drawing the altitudes of the circles down to $BC$ $a$ and $b$. Now we have the length of side $BC$ of being $(2)(2022)+1+1+a+b$. However, the side $BC$ can also be written as $(6)(68)+34+34+34a+34b$, due to similar triangles from the second diagram. If we set the equations equal, we have $\frac{1190}{11} = a+b$. Call the radius of the incircle $r$, then we have the side BC to be $r(a+b)$. We find $r$ as $\frac{4046+\frac{1190}{11}}{\frac{1190}{11}}$, which simplifies to $\frac{10+((34)(11))}{10}$,so we have $\frac{192}{5}$, which sums to $\boxed{197}$.

Solution 2

Assume that $ABC$ is isosceles with $AB=AC$.

If we let $P_1$ be the intersection of $BC$ and the leftmost of the eight circles of radius $34$, $N_1$ the center of the leftmost circle, and $M_1$ the intersection of the leftmost circle and $AB$, and we do the same for the $2024$ circles of radius $1$, naming the points $P_2$, $N_2$, and $M_2$, respectively, then we see that $BP_1N_1M_1\sim BP_2N_2M_2$. The same goes for vertex $C$, and the corresponding quadrilaterals are congruent.

Let $x=BP_2$. We see that $BP_1=34x$ by similarity ratios (due to the radii). The corresponding figures on vertex $C$ are also these values. If we combine the distances of the figures, we see that $BC=2x+4046$ and $BC=68x+476$, and solving this system, we find that $x=\frac{595}{11}$.

If we consider that the incircle of $\triangle ABC$ is essentially the case of $1$ circle with $r$ radius (the inradius of $\triangle ABC$, we can find that $BC=2rx$. From $BC=2x+4046$, we have:

$r=1+\frac{2023}{x}$

$=1+\frac{11\cdot2023}{595}$

$=1+\frac{187}{5}$

$=\frac{192}{5}$

Thus the answer is $192+5=\boxed{197}$.

~eevee9406

Solution 3

Let $x = \cot{\frac{B}{2}} + \cot{\frac{C}{2}}$. By representing $BC$ in two ways, we have the following: \[34x + 7\cdot 34\cdot 2 = BC\] \[x + 2023 \cdot 2 = BC\]

Solving we find $x = \frac{1190}{11}$. Now draw the inradius, let it be $r$. We find that $rx =BC$, hence \[xr = x + 4046 \implies r-1 = \frac{11}{1190}\cdot 4046 = \frac{187}{5}.\] Thus \[r = \frac{192}{5} \implies \boxed{197}.\] ~AtharvNaphade

Solution 4

First, let the circle tangent to $AB$ and $BC$ be $O$ and the other circle that is tangent to $AC$ and $BC$ be $R$. Let $x$ be the distance from the tangency point on line segment $BC$ of the circle $O$ to $B$. Also, let $y$ be the distance of the tangency point of circle $R$ on the line segment $BC$ to point $C$. Realize that we can let $n$ be the number of circles tangent to line segment $BC$ and $r$ be the corresponding radius of each of the circles. Also, the circles that are tangent to $BC$ are similar. So, we can build the equation $BC = (x+y+2(n-1)) \times r$. Looking at the given information, we see that when $n=8$, $r=34$, and when $n=2024$, $r=1$, and we also want to find the radius $r$ in the case where $n=1$. Using these facts, we can write the following equations:

$BC = (x+y+2(8-1)) \times 34 = (x+y+2(2024-1)) \times 1 = (x+y+2(1-1)) \times r$

We can find that $x+y = \frac{1190}{11}$ . Now, let $(x+y+2(2024-1)) \times 1 = (x+y+2(1-1)) \times r$.

Substituting $x+y = \frac{1190}{11}$ in, we find that \[r = \frac{192}{5} \implies \boxed{197}.\]

~EaZ_Shadow

Solution 5 (one variable)

Define $I, x_1, x_8, y_1, y_{2024}$ to be the incenter and centers of the first and last circles of the $8$ and $2024$ tangent circles to $BC,$ and define $r$ to be the inradius of triangle $\bigtriangleup ABC.$ We calculate $\overline{x_1x_8} = 34 \cdot 14$ and $\overline{y_1y_{2024}} = 1 \cdot 4046$ because connecting the center of the circles voids two extra radii.


We can easily see that $B, x_1, x_8,$ and $I$ are collinear, and the same follows for $C, y_1, y_2024,$ and $I$ (think angle bisectors).


We observe that triangles $\bigtriangleup I x_1 x_8$ and $\bigtriangleup I y_1 y_{2024}$ are similar, and therefore the ratio of the altitude to the base is the same, so we note

\[\frac{\text{altitude}}{\text{base}} = \frac{r-34}{34\cdot 14} = \frac{r-1}{1\cdot 4046}.\]


Solving yields $r = \frac{192}{5},$ so the answer is $192+5 = \boxed{197}.$

-spectraldragon8

Video Solution（Chinese）subtitle in English

https://youtu.be/q8N-zzlUFpA

Video Solution 1 by OmegaLearn.org

https://youtu.be/MWTf6Jr8UwU

Video Solution

https://youtu.be/qlrVguS79GA

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution

https://www.youtube.com/watch?v=MWhLgPr-ZR8&t=716s&ab_channel=TheBeautyofMath (beautyofmath)

See also

2024 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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