Difference between revisions of "2019 AMC 12A Problems/Problem 25"

Latest revision as of 23:22, 3 November 2024

Problem

Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\circ$ , $60^\circ$ , and $60.001^\circ$ . For each positive integer $n$ , define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$ . Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$ , and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$ . What is the least positive integer $n$ for which $\triangle A_nB_nC_n$ is obtuse?

$\textbf{(A) } 10 \qquad \textbf{(B) }11 \qquad \textbf{(C) } 13\qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15$

Solution 1

For all nonnegative integers $n$ , let $\angle C_nA_nB_n=x_n$ , $\angle A_nB_nC_n=y_n$ , and $\angle B_nC_nA_n=z_n$ .

Note that quadrilateral $A_0B_0A_1B_1$ is cyclic since $\angle A_0A_1B_0=\angle A_0B_1B_0=90^\circ$ ; thus, $\angle A_0A_1B_1=\angle A_0B_0B_1=90^\circ-x_0$ . By a similar argument, $\angle A_0A_1C_1=\angle A_0C_0C_1=90^\circ-x_0$ . Thus, $x_1=\angle A_0A_1B_1+\angle A_0A_1C_1=180^\circ-2x_0$ . By a similar argument, $y_1=180^\circ-2y_0$ and $z_1=180^\circ-2z_0$ .

Therefore, for any positive integer $n$ , we have $x_n=180^\circ-2x_{n-1}$ (identical recurrence relations can be derived for $y_n$ and $z_n$ ). To find an explicit form for this recurrence, we guess that the constant term is related exponentially to $n$ (and the coefficient of $x_0$ is $(-2)^n$ ). Hence, we let $x_n=pq^n+r+(-2)^nx_0$ . We will solve for $p$ , $q$ , and $r$ by iterating the recurrence to obtain $x_1=180^\circ-2x_0$ , $x_2=4x_0-180^\circ$ , and $x_3=540-8x_0$ . Letting $n=1,2,3$ respectively, we have $\begin{align} pq+r&=180 \\ pq^2+r&=-180 \\ pq^3+r&=540 \end{align}$

Subtracting $(1)$ from $(3)$ , we have $pq(q^2-1)=360$ , and subtracting $(1)$ from $(2)$ gives $pq(q-1)=-360$ . Dividing these two equations gives $q+1=-1$ , so $q=-2$ . Substituting back, we get $p=-60$ and $r=60$ .

We will now prove that for all positive integers $n$ , $x_n=-60(-2)^n+60+(-2)^nx_0=(-2)^n(x_0-60)+60$ via induction. Clearly the base case of $n=1$ holds, so it is left to prove that $x_{n+1}=(-2)^{n+1}(x_0-60)+60$ assuming our inductive hypothesis holds for $n$ . Using the recurrence relation, we have $\begin{align*} x_{n+1}&=180-2x_n \\ &=180-2((-2)^n(x_0-60)+60) \\ &=(-2)^{n+1}(x_0-60)+60 \end{align*}$

Our induction is complete, so for all positive integers $n$ , $x_n=(-2)^n(x_0-60)+60$ . Identical equalities hold for $y_n$ and $z_n$ .

The problem asks for the smallest $n$ such that either $x_n$ , $y_n$ , or $z_n$ is greater than $90^\circ$ . WLOG, let $x_0=60^\circ$ , $y_0=59.999^\circ$ , and $z_0=60.001^\circ$ . Thus, $x_n=60^\circ$ for all $n$ , $y_n=-(-2)^n(0.001)+60$ , and $z_n=(-2)^n(0.001)+60$ . Solving for the smallest possible value of $n$ in each sequence, we find that $n=15$ gives $y_n>90^\circ$ . Therefore, the answer is $\boxed{\textbf{(E) } 15}$ .

Solution 2

We start from Solution 1 until we reach the recurrence relation $x_n = 180 - 2x_{n - 1}.$ Iterate this again, to get $x_{n - 1} = 180 - 2x_{n - 2}.$ Subtract the two, getting $x_{n} = -x_{n - 1} + 2x_{n - 2}.$ This recurrence has characteristic equation $x^2 + x - 2 = 0 = (x + 2)(x - 1) = 0 \iff x = -2, 1.$ Now, write $x_n = p + q \cdot (-2)^n.$ We obtain similar recursions for $y, z$ that can be easily solved by getting $x_1, y_1, z_1$ from the original recursive formula and then using those two values to solve for $p$ and $q.$ Then proceed with the last paragraph of Solution 1.

Solution 3 (If you're short on time)

We note that the problem seems quite complicated, but since it is an AMC 12, the difference between the largest angle of $\triangle A_nB_nC_n$ and $60^{\circ}$ (we call this quantity S) most likely reduces to a simpler problem like some repeating sequence. The only obvious sequence (for the answer choices) is a geometric sequence with an integer common ratio. From here, we get that $S_n=0.001S^n$ and we need $S_n>30$ . The first power of two greater than $30,000$ is $2^{15}$ thus our answer is $\boxed{\textbf{(E) } 15}$ .

~Dhillonr25

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2019amc12a/497

~ dolphin7

@@ Line 1: / Line 1: @@
 ==Problem==
-Let <math>\triangle A_0B_0C_0</math> be a triangle whose angle measures are exactly <math>59.999^\circ</math>, <math>60^\circ</math>, and <math>60.001^\circ</math>. For each positive integer <math>n</math> define <math>A_n</math> to be the foot of the altitude from <math>A_{n-1}</math> to line <math>B_{n-1}C_{n-1}</math>. Likewise, define <math>B_n</math> to be the foot of the altitude from <math>B_{n-1}</math> to line <math>A_{n-1}C_{n-1}</math>, and <math>C_n</math> to be the foot of the altitude from <math>C_{n-1}</math> to line <math>A_{n-1}B_{n-1}</math>. What is the least positive integer <math>n</math> for which <math>\triangle A_nB_nC_n</math> is obtuse?
+Let <math>\triangle A_0B_0C_0</math> be a triangle whose angle measures are exactly <math>59.999^\circ</math>, <math>60^\circ</math>, and <math>60.001^\circ</math>. For each positive integer <math>n</math>, define <math>A_n</math> to be the foot of the altitude from <math>A_{n-1}</math> to line <math>B_{n-1}C_{n-1}</math>. Likewise, define <math>B_n</math> to be the foot of the altitude from <math>B_{n-1}</math> to line <math>A_{n-1}C_{n-1}</math>, and <math>C_n</math> to be the foot of the altitude from <math>C_{n-1}</math> to line <math>A_{n-1}B_{n-1}</math>. What is the least positive integer <math>n</math> for which <math>\triangle A_nB_nC_n</math> is obtuse?
-<math>\phantom{}</math>
 <math>\textbf{(A) } 10 \qquad \textbf{(B) }11 \qquad \textbf{(C) } 13\qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15</math>
-==Solution==
+==Solution 1==
 For all nonnegative integers <math>n</math>, let <math>\angle C_nA_nB_n=x_n</math>, <math>\angle A_nB_nC_n=y_n</math>, and <math>\angle B_nC_nA_n=z_n</math>.
-Note that quadrilateral <math>A_0B_0A_1B_1</math> is cyclic since <math>\angle A_0A_1B_0=\angle A_0B_1B_0=90^\circ</math>; thus, <math>\angle A_0A_1B_1=\angle A_0B_0B_1=90^\circ-x_0</math>. By a similar argument, <math>\angle A_0A_1C_1=\angle A_0C_0C_1=90^\circ-x_0</math>. Thus, <math>x_1=\angle A_0A_1B_1+\angle A_0A_1C_1=180^\circ-2x_0</math>. By a symmetric argument, <math>y_1=180^\circ-2y_0</math> and <math>z_1=180^\circ-2z_0</math>.
+Note that quadrilateral <math>A_0B_0A_1B_1</math> is cyclic since <math>\angle A_0A_1B_0=\angle A_0B_1B_0=90^\circ</math>; thus, <math>\angle A_0A_1B_1=\angle A_0B_0B_1=90^\circ-x_0</math>. By a similar argument, <math>\angle A_0A_1C_1=\angle A_0C_0C_1=90^\circ-x_0</math>. Thus, <math>x_1=\angle A_0A_1B_1+\angle A_0A_1C_1=180^\circ-2x_0</math>. By a similar argument, <math>y_1=180^\circ-2y_0</math> and <math>z_1=180^\circ-2z_0</math>.
 Therefore, for any positive integer <math>n</math>, we have <math>x_n=180^\circ-2x_{n-1}</math> (identical recurrence relations can be derived for <math>y_n</math> and <math>z_n</math>). To find an explicit form for this recurrence, we guess that the constant term is related exponentially to <math>n</math> (and the coefficient of <math>x_0</math> is <math>(-2)^n</math>). Hence, we let <math>x_n=pq^n+r+(-2)^nx_0</math>. We will solve for <math>p</math>, <math>q</math>, and <math>r</math> by iterating the recurrence to obtain <math>x_1=180^\circ-2x_0</math>, <math>x_2=4x_0-180^\circ</math>, and <math>x_3=540-8x_0</math>. Letting <math>n=1,2,3</math> respectively, we have <cmath>\begin{align}
@@ Line 17: / Line 16: @@
 \end{align}</cmath>
-Subtracting <math>(1)</math> from <math>(3)</math>, we have <math>pq(q^2-1)=360</math>, and subtracting <math>(1)</math> from <math>(2)</math> gives <math>pq(q-1)=-360</math>. Dividing these two equations gives <math>q+1=-1</math>, so <math>q=-2</math>. Substituting back, we get <math>p=-60</math> and <math>r=60</math>. Hence, for all positive integers <math>n</math>, <math>x_n=-60(-2)^n+60+(-2)^nx_0=(-2)^n(x_0-60)+60</math>, and the same holds for <math>y_n</math> and <math>z_n</math>.
+Subtracting <math>(1)</math> from <math>(3)</math>, we have <math>pq(q^2-1)=360</math>, and subtracting <math>(1)</math> from <math>(2)</math> gives <math>pq(q-1)=-360</math>. Dividing these two equations gives <math>q+1=-1</math>, so <math>q=-2</math>. Substituting back, we get <math>p=-60</math> and <math>r=60</math>.
-The problem asks for the smallest <math>n</math> such that either <math>x_n</math>, <math>y_n</math>, or <math>z_n</math> is greater than <math>90^\circ</math>. WLOG, let <math>x_0=60^\circ</math>, <math>y_0=59.999^\circ</math>, and <math>z_0=60.001^\circ</math>. Thus, <math>x_n=60^\circ</math> for all <math>n</math>, <math>y_n=-(-2)^n(0.001)+60</math>, and <math>z_n=(-2)^n(0.001)+60</math>. Solving for the smallest possible value of <math>n</math> in each sequence, we find that <math>n=15</math> gives <math>y_n>90^\circ</math>. Therefore, the answer is <math>\boxed{\bold (E) 15}}</math>.
+We will now prove that for all positive integers <math>n</math>, <math>x_n=-60(-2)^n+60+(-2)^nx_0=(-2)^n(x_0-60)+60</math> via induction. Clearly the base case of <math>n=1</math> holds, so it is left to prove that <math>x_{n+1}=(-2)^{n+1}(x_0-60)+60</math> assuming our inductive hypothesis holds for <math>n</math>. Using the recurrence relation, we have <cmath>\begin{align*}
+x_{n+1}&=180-2x_n \\
+&=180-2((-2)^n(x_0-60)+60) \\
+&=(-2)^{n+1}(x_0-60)+60
+\end{align*}</cmath>
+Our induction is complete, so for all positive integers <math>n</math>, <math>x_n=(-2)^n(x_0-60)+60</math>. Identical equalities hold for <math>y_n</math> and <math>z_n</math>.
+The problem asks for the smallest <math>n</math> such that either <math>x_n</math>, <math>y_n</math>, or <math>z_n</math> is greater than <math>90^\circ</math>. WLOG, let <math>x_0=60^\circ</math>, <math>y_0=59.999^\circ</math>, and <math>z_0=60.001^\circ</math>. Thus, <math>x_n=60^\circ</math> for all <math>n</math>, <math>y_n=-(-2)^n(0.001)+60</math>, and <math>z_n=(-2)^n(0.001)+60</math>. Solving for the smallest possible value of <math>n</math> in each sequence, we find that <math>n=15</math> gives <math>y_n>90^\circ</math>. Therefore, the answer is <math>\boxed{\textbf{(E) } 15}</math>.
+==Solution 2==
+We start from Solution 1 until we reach the recurrence relation <math>x_n = 180 - 2x_{n - 1}.</math> Iterate this again, to get <math>x_{n - 1} = 180 - 2x_{n - 2}.</math> Subtract the two, getting <math>x_{n} = -x_{n - 1} + 2x_{n - 2}.</math> This recurrence has characteristic equation <math>x^2 + x - 2 = 0 = (x + 2)(x - 1) = 0 \iff x = -2, 1.</math> Now, write <cmath>x_n = p + q \cdot (-2)^n.</cmath> We obtain similar recursions for <math>y, z</math> that can be easily solved by getting <math>x_1, y_1, z_1</math> from the original recursive formula and then using those two values to solve for <math>p</math> and <math>q.</math> Then proceed with the last paragraph of Solution 1.
+==Solution 3 (If you're short on time)==
+We note that the problem seems quite complicated, but since it is an AMC 12, the difference between the largest angle of <math>\triangle A_nB_nC_n</math> and <math>60^{\circ}</math> (we call this quantity S) most likely reduces to a simpler problem like some repeating sequence. The only obvious sequence (for the answer choices) is a geometric sequence with an integer common ratio. From here, we get that <math>S_n=0.001S^n</math> and we need <math>S_n>30</math>. The first power of two greater than <math>30,000</math> is <math>2^{15}</math> thus our answer is <math>\boxed{\textbf{(E) } 15}</math>.
+~Dhillonr25
+== Video Solution by Richard Rusczyk ==
+https://artofproblemsolving.com/videos/amc/2019amc12a/497
+~ dolphin7
 ==See Also==

2019 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 24	Followed by Last Problem
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