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

Revision as of 16:16, 3 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 $m+n$ .

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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 $x = \cot{\frac{A}{2}} + \cot{\frac{B}{2}}$ . 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 the 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 these 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 $r = \frac{192}{5} \implies \boxed{197}.$

~Rainier2020

Video Solution 1 by OmegaLearn.org

https://youtu.be/MWTf6Jr8UwU

@@ Line 39: / Line 39: @@
 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 the 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 these 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>
+~Rainier2020

2024 AIME I (Problems • Answer Key • Resources)
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