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Difference between revisions of "2024 AIME I Problems/Problem 8"

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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>.
  
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==Solution==
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==Solution 1==
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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 <math>a</math> and <math>b</math>. Now we have the length of side BC of being <math>(2)(2022)+1+1+a+b</math>. However, the side BC 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 r, 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 197.
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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>.
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==Solution 2==
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Assume that <math>ABC</math> is isosceles with <math>AB=AC</math>.
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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.
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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>.
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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:
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<math>r=1+\frac{2023}{x}</math>
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<math>=1+\frac{11\cdot2023}{595}</math>
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<math>=1+\frac{187}{5}</math>
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<math>=\frac{192}{5}</math>
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Thus the answer is <math>192+5=\boxed{197}</math>.
  
 
==See also==
 
==See also==

Revision as of 17:56, 2 February 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 $m+n$.

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 $192+5=\boxed{197}$.

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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