Difference between revisions of "2011 AMC 10B Problems/Problem 25"

(Solution 2)
(Solution 2)
 
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== Problem ==
 
== Problem ==
  
Let <math>T_1</math> be a triangle with sides <math>2011, 2012,</math> and <math>2013</math>. For <math>n \ge 1</math>, if <math>T_n = \triangle ABC</math> and <math>D, E,</math> and <math>F</math> are the points of tangency of the incircle of <math>\triangle ABC</math> to the sides <math>AB, BC</math> and <math>AC,</math> respectively, then <math>T_{n+1}</math> is a triangle with side lengths <math>AD, BE,</math> and <math>CF,</math> if it exists. What is the perimeter of the last triangle in the sequence <math>( T_n )</math>?
+
Let <math>T_1</math> be a triangle with side lengths <math>2011, 2012,</math> and <math>2013</math>. For <math>n \ge 1</math>, if <math>T_n = \triangle ABC</math> and <math>D, E,</math> and <math>F</math> are the points of tangency of the incircle of <math>\triangle ABC</math> to the sides <math>AB, BC</math>, and <math>AC,</math> respectively, then <math>T_{n+1}</math> is a triangle with side lengths <math>AD, BE,</math> and <math>CF,</math> if it exists. What is the perimeter of the last triangle in the sequence <math>( T_n )</math>?
  
 
<math> \textbf{(A)}\ \frac{1509}{8} \qquad\textbf{(B)}\ \frac{1509}{32} \qquad\textbf{(C)}\ \frac{1509}{64} \qquad\textbf{(D)}\ \frac{1509}{128} \qquad\textbf{(E)}\ \frac{1509}{256}</math>
 
<math> \textbf{(A)}\ \frac{1509}{8} \qquad\textbf{(B)}\ \frac{1509}{32} \qquad\textbf{(C)}\ \frac{1509}{64} \qquad\textbf{(D)}\ \frac{1509}{128} \qquad\textbf{(E)}\ \frac{1509}{256}</math>
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Solving gives:
 
Solving gives:
  
<math>x= \frac{a-1}{2} </math>
+
<math>x= \frac{a}{2} -1</math>
  
 
<math>y = \frac{a}{2}</math>
 
<math>y = \frac{a}{2}</math>
  
<math>z = \frac{a+1}{2}</math>
+
<math>z = \frac{a}{2}+1</math>
  
 
Subbing in gives that <math>T_2</math> has sides of <math>1005, 1006, 1007</math>.
 
Subbing in gives that <math>T_2</math> has sides of <math>1005, 1006, 1007</math>.
  
<math>T_3</math> can easily be derivied from this as the sides still differ by 1 hence the above solutions still work (now with <math>a=1006</math>). All additional triangles will differ by one as the solutions above differ by one so this process can be repeated indefinitely until the side lengths no longer form a triangle.
+
<math>T_3</math> can easily be derived from this as the sides still differ by 1 hence the above solutions still work (now with <math>a=1006</math>). All additional triangles will differ by one as the solutions above differ by one so this process can be repeated indefinitely until the side lengths no longer form a triangle.
  
 
Subbing in gives <math>T_3</math> with sides <math>502, 503, 504</math>.
 
Subbing in gives <math>T_3</math> with sides <math>502, 503, 504</math>.
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<math>T_{11}</math> would have sides <math>\frac{247}{256}, \frac{503}{256}, \frac{759}{256}</math> but these lengths do not make a  
 
<math>T_{11}</math> would have sides <math>\frac{247}{256}, \frac{503}{256}, \frac{759}{256}</math> but these lengths do not make a  
triangle as <math>\frac{247}{256} + \frac{503}{256} < \frac{759}{256}</math>.
+
triangle as  
 +
<cmath>\frac{247}{256} + \frac{503}{256} < \frac{759}{256}</cmath>
  
  
 
Likewise, you could create an equation instead of listing all the triangles to <math>T_{11}</math>.
 
Likewise, you could create an equation instead of listing all the triangles to <math>T_{11}</math>.
The sides of a triangle <math>T_{k}</math> would be <math>\frac{503}{2^{k-3}} - 1, \frac{503}{2^{k-3}}, \frac{503}{2^{k-3}} + 1</math>.
+
The sides of a triangle <math>T_{k}</math> would be  
We then have <math>503 - 2^{k-3} + 503 > 503 + 2^{k-3} \rightarrow 1006 - 2^{k-3} > 503 + 2^{k-3} \rightarrow 503 > 2^{k-2} \rightarrow 9 > k-2 \rightarrow k < 11</math>.
+
<cmath>\frac{503}{2^{k-3}} - 1, \frac{503}{2^{k-3}}, \frac{503}{2^{k-3}} + 1</cmath>
 +
We then have  
 +
<cmath>503 - 2^{k-3} + 503 > 503 + 2^{k-3}</cmath>
 +
<cmath>1006 - 2^{k-3} > 503 + 2^{k-3}</cmath>
 +
<cmath>503 > 2^{k-2}</cmath>
 +
<cmath>9 > k-2</cmath>
 +
<cmath>k < 11</cmath>
 
Hence, the first triangle which does not exist in this sequence is <math>T_{11}</math>.
 
Hence, the first triangle which does not exist in this sequence is <math>T_{11}</math>.
  
 +
Hence the perimeter is
 +
<cmath>\frac{375}{128} + \frac{503}{128} + \frac{631}{128} = \boxed{\textbf{(D)} \frac{1509}{128}}</cmath>.
  
Hence the perimeter is <math>\frac{375}{128} + \frac{503}{128} + \frac{631}{128} = \boxed{\textbf{(D)} \frac{1509}{128}}</math>
+
==Solution 2==
  
==Solution 2 ==
+
Proceeding similarly to the first solution, we have that sides of each triangle are of the form <math>a, a+1, a+2</math> for some number <math>a</math>. Also, note that the perimeter of each triangle is half of the previous one. In order for the triangle to not exist, it must not satisfy the triangle inequality, meaning that <math>a + a + 1 < a+2 \Rightarrow a<1</math>. Then, the perimeter would be <math>a + a + 1 + a + 2 = 3a + 3 < 6</math>. So, to have a proper triangle, we have <math>\frac{3018}{2^{k}} > 6 \Rightarrow 2^k < 503 \Rightarrow 2^{k} < 512</math>. The first triangle to not work would have perimeter <math>\frac{3018}{256} = \frac{1509}{128}</math>,  thus the answer is  <math>\boxed{\textbf{(D)} \frac{1509}{128}}</math>.
  
Proceeding similarly to the first solution, we have that sides of each triangle are of the form <math>a, a+1, a+2</math> for some number <math>a</math>. Also, note that the perimeter of each triangle is half of the previous one. In order for the triangle to not exist, it must not satisfy the triangle inequality, meaning that <math>a + a + 1 < a+2 \Rightarrow a<1</math>. Then, the perimeter would be <math>a + a + 1 + a + 2 = 3a + 3 < 6</math>. So, we have <math>\frac{3018}{2^{k}} > 6 \Rightarrow 2^k < 503 \Rightarrow 2^{k} \leq 512</math>. The first triangle to not work would have perimeter <math>\frac{3018}{512} = \frac{1509}{256} = 5.89453125</math>, thus the answer is <math>\boxed{\textbf{(D)} \frac{1509}{128}}</math>.
+
==Solution 3==
 +
This problem is a PoP(Power of Point) Bash exercise. What I did to solve this problem was look at recursive perimeters. The first ever perimeter is <math>P_1 = 2011 + 2012 + 2013</math>. Next, by PoP, inscribing the circle gives us three new lengths, namely <math>AD, BE, CF</math>. Denote <math>AD</math> as <math>x_1</math> and homogeneously the others. Then, <math>x_1 + x_2 = 2011, x_2 + x_3 = 2012</math>, and <math>x_1 + x_3 = 2013</math>. If we add all these equations up and divide by <math>2</math>, we get <math>x_1 + x_2 + x_3 = \frac{2011 + 2012 + 2013}{2}</math>. Writing this with <math>P_1 = 2011 + 2012 + 2013</math>, we get that our new perimeter, <math>P_2</math>, is indeed equal to <math>\frac{P_1}{2}</math>. Similarly, by the same concept, we get that <math>P_3 = \frac{P_1}{4}</math> and the pattern keeps going. In general, I found that for each new perimeter <math>P_n</math>, <math>P_n = \frac{P_1}{2^{n-1}}</math>. Now, substituting in the numerical value of <math>P_1</math>, we get that <math>P_n = \frac{2011 + 2012 + 2013}{2^{n-1}}</math>. If you keep dividing the numerator and denominator by 2, I got that: <math>P_n = \frac{1509}{2^{n-3}}</math>. This representation of a new perimeter in terms of <math>n</math> looks very similar to the option choices, so we're on the right path. Now, all we need to do is find out the last value of <math>n</math> when the sequence stops working. By the Triangular Inequality, we may be able to finish this off. Now, I won't go into depths here, but listing out terms and using the Triangular Inequality gives us:
 +
First Sequence:
 +
<math>x_1 < 2012</math>
 +
 
 +
<math>x_2 < 2013</math>
 +
 
 +
<math>x_3 < 2011</math>.
 +
The ordered triple for this set of <math>(x_1, x_2, x_3)</math> is <math>(1005, 1006, 1007)</math> if you solve the PoP system of equations we got earlier. If you keep listing out terms, we can come up with the general form, and that is:
 +
<math>x_1 < \frac{503}{2^{n-4}}</math>
 +
 
 +
<math>x_2 < \frac{503}{2^{n-4}} + 1</math>
 +
 
 +
<math>x_3 < \frac{503}{2^{n-4}} - 1</math>.
 +
The ordered triple for this set of <math>(x_1, x_2, x_3)</math> is <math>(\frac{503}{2^{n-4}}, \frac{503}{2^{n-4}} - 1, \frac{503}{2^{n-4}} + 1)</math>. Now, if we plug in these values of <math>x_1, x_2</math>, and <math>x_3</math> into the inequalities, we see that the first two are always satisfied, but the last one is only satisfied when:
 +
<math>2^{n-2} < 503</math>(You will get this if you simplify the last inequality). The last <math>n</math> for this to be satisfied is when <math>n = 10</math>. If we go up to our general representation of <math>P_n</math>, we see that <math>P_n = \frac{1509}{2^{n-3}}</math>. Plugging in <math>10</math> because this is the last <math>n</math>(and also the last triangle), we our final answer of <math>\frac{1509}{2^{7}} = \boxed{\frac{1509}{128}}</math> or <math>\boxed{D}</math>.
 +
 
 +
~ilikemath247365
  
 
==See Also==
 
==See Also==
 
+
Identical problem to the [[2011 AMC 12B Problems/Problem 22]].
  
 
{{AMC10 box|year=2011|ab=B|num-b=24|after=Last Problem}}
 
{{AMC10 box|year=2011|ab=B|num-b=24|after=Last Problem}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 15:38, 17 February 2025

Problem

Let $T_1$ be a triangle with side lengths $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n = \triangle ABC$ and $D, E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB, BC$, and $AC,$ respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE,$ and $CF,$ if it exists. What is the perimeter of the last triangle in the sequence $( T_n )$?

$\textbf{(A)}\ \frac{1509}{8} \qquad\textbf{(B)}\ \frac{1509}{32} \qquad\textbf{(C)}\ \frac{1509}{64} \qquad\textbf{(D)}\ \frac{1509}{128} \qquad\textbf{(E)}\ \frac{1509}{256}$

Solution 1

By constructing the bisectors of each angle and the perpendicular radii of the incircle the triangle consists of 3 kites.

File2011AMC10B25.png

Hence $AD=AF$ and $BD=BE$ and $CE=CF$. Let $AD = x, BD = y$ and $CE = z$ gives three equations:

$x+y = a-1$

$x+z = a$

$y+z = a+1$

(where $a = 2012$ for the first triangle.)

Solving gives:

$x= \frac{a}{2} -1$

$y = \frac{a}{2}$

$z = \frac{a}{2}+1$

Subbing in gives that $T_2$ has sides of $1005, 1006, 1007$.

$T_3$ can easily be derived from this as the sides still differ by 1 hence the above solutions still work (now with $a=1006$). All additional triangles will differ by one as the solutions above differ by one so this process can be repeated indefinitely until the side lengths no longer form a triangle.

Subbing in gives $T_3$ with sides $502, 503, 504$.

$T_4$ has sides $\frac{501}{2}, \frac{503}{2}, \frac{505}{2}$.

$T_5$ has sides $\frac{499}{4}, \frac{503}{4}, \frac{507}{4}$.

$T_6$ has sides $\frac{495}{8}, \frac{503}{8}, \frac{511}{8}$.

$T_7$ has sides $\frac{487}{16}, \frac{503}{16}, \frac{519}{16}$.

$T_8$ has sides $\frac{471}{32}, \frac{503}{32}, \frac{535}{32}$.

$T_9$ has sides $\frac{439}{64}, \frac{503}{64}, \frac{567}{64}$.

$T_{10}$ has sides $\frac{375}{128}, \frac{503}{128}, \frac{631}{128}$.

$T_{11}$ would have sides $\frac{247}{256}, \frac{503}{256}, \frac{759}{256}$ but these lengths do not make a triangle as \[\frac{247}{256} + \frac{503}{256} < \frac{759}{256}\]


Likewise, you could create an equation instead of listing all the triangles to $T_{11}$. The sides of a triangle $T_{k}$ would be \[\frac{503}{2^{k-3}} - 1, \frac{503}{2^{k-3}}, \frac{503}{2^{k-3}} + 1\] We then have \[503 - 2^{k-3} + 503 > 503 + 2^{k-3}\] \[1006 - 2^{k-3} > 503 + 2^{k-3}\] \[503 > 2^{k-2}\] \[9 > k-2\] \[k < 11\] Hence, the first triangle which does not exist in this sequence is $T_{11}$.

Hence the perimeter is \[\frac{375}{128} + \frac{503}{128} + \frac{631}{128} = \boxed{\textbf{(D)} \frac{1509}{128}}\].

Solution 2

Proceeding similarly to the first solution, we have that sides of each triangle are of the form $a, a+1, a+2$ for some number $a$. Also, note that the perimeter of each triangle is half of the previous one. In order for the triangle to not exist, it must not satisfy the triangle inequality, meaning that $a + a + 1 < a+2 \Rightarrow a<1$. Then, the perimeter would be $a + a + 1 + a + 2 = 3a + 3 < 6$. So, to have a proper triangle, we have $\frac{3018}{2^{k}} > 6 \Rightarrow 2^k < 503 \Rightarrow 2^{k} < 512$. The first triangle to not work would have perimeter $\frac{3018}{256} = \frac{1509}{128}$, thus the answer is $\boxed{\textbf{(D)} \frac{1509}{128}}$.

Solution 3

This problem is a PoP(Power of Point) Bash exercise. What I did to solve this problem was look at recursive perimeters. The first ever perimeter is $P_1 = 2011 + 2012 + 2013$. Next, by PoP, inscribing the circle gives us three new lengths, namely $AD, BE, CF$. Denote $AD$ as $x_1$ and homogeneously the others. Then, $x_1 + x_2 = 2011, x_2 + x_3 = 2012$, and $x_1 + x_3 = 2013$. If we add all these equations up and divide by $2$, we get $x_1 + x_2 + x_3 = \frac{2011 + 2012 + 2013}{2}$. Writing this with $P_1 = 2011 + 2012 + 2013$, we get that our new perimeter, $P_2$, is indeed equal to $\frac{P_1}{2}$. Similarly, by the same concept, we get that $P_3 = \frac{P_1}{4}$ and the pattern keeps going. In general, I found that for each new perimeter $P_n$, $P_n = \frac{P_1}{2^{n-1}}$. Now, substituting in the numerical value of $P_1$, we get that $P_n = \frac{2011 + 2012 + 2013}{2^{n-1}}$. If you keep dividing the numerator and denominator by 2, I got that: $P_n = \frac{1509}{2^{n-3}}$. This representation of a new perimeter in terms of $n$ looks very similar to the option choices, so we're on the right path. Now, all we need to do is find out the last value of $n$ when the sequence stops working. By the Triangular Inequality, we may be able to finish this off. Now, I won't go into depths here, but listing out terms and using the Triangular Inequality gives us: First Sequence: $x_1 < 2012$

$x_2 < 2013$

$x_3 < 2011$. The ordered triple for this set of $(x_1, x_2, x_3)$ is $(1005, 1006, 1007)$ if you solve the PoP system of equations we got earlier. If you keep listing out terms, we can come up with the general form, and that is: $x_1 < \frac{503}{2^{n-4}}$

$x_2 < \frac{503}{2^{n-4}} + 1$

$x_3 < \frac{503}{2^{n-4}} - 1$. The ordered triple for this set of $(x_1, x_2, x_3)$ is $(\frac{503}{2^{n-4}}, \frac{503}{2^{n-4}} - 1, \frac{503}{2^{n-4}} + 1)$. Now, if we plug in these values of $x_1, x_2$, and $x_3$ into the inequalities, we see that the first two are always satisfied, but the last one is only satisfied when: $2^{n-2} < 503$(You will get this if you simplify the last inequality). The last $n$ for this to be satisfied is when $n = 10$. If we go up to our general representation of $P_n$, we see that $P_n = \frac{1509}{2^{n-3}}$. Plugging in $10$ because this is the last $n$(and also the last triangle), we our final answer of $\frac{1509}{2^{7}} = \boxed{\frac{1509}{128}}$ or $\boxed{D}$.

~ilikemath247365

See Also

Identical problem to the 2011 AMC 12B Problems/Problem 22.

2011 AMC 10B (ProblemsAnswer KeyResources)
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Problem 24
Followed by
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