Difference between revisions of "2011 AMC 12A Problems/Problem 23"

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Problem

Let $f(z)= \frac{z+a}{z+b}$ and $g(z)=f(f(z))$ , where $a$ and $b$ are complex numbers. Suppose that $\left| a \right| = 1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $\left| b \right|$ ?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \sqrt{2}-1 \qquad \textbf{(C)}\ \sqrt{3}-1 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

Solution

Solution 1

Lemma) if $f(z) = \frac{z + a}{z + b}$ , then $f(f(f(f(\frac{-a(1+b)}{a+1})))) = \frac{a ( 1 + b)}{ a+b^2}$

$f(f(f(f(\frac{-a(1+b)}{a+1})))) = f(f(f(-a))) = f(f(0)) = f(\frac{a}{b}) = \frac{a + ab}{a + b^2}$

Well, let us consider the cases where each of those step is definite ( $f(-b)$ is never evaluate).

So, we have $\frac{-a(1+b)}{a+1} \neq - b$ ,

$-a - ab \neq - ab - b$

$a \neq b$ --- (exception -> case 2)

$-a \neq -b$ --- (exception -> case 3)

$0 \neq -b$ --- (exception -> case 4)

$\frac{a}{b} \neq -b$

$a \neq -b ^2$ --- (exception -> case 5)

If it is not any of the above 5 cases, then

$\frac{-a(1+b)}{a+1} = \frac{a + ab}{a + b^2}$

if $a(1+b) \neq 0$ (--- exception -> case 6), then $-(a+b^2) = a+1$ , $b^2 = -2a - 1$ , $1 \le |b^2| \le 3$

Hence, it is possible maximum of $|b| = \sqrt{3}$ and minimum is 1.

2 possible combination of $(a,b)$ are $(1, 3i)$ and $(-1, 1)$ . Case 2) $a = b$ , then $|b| = 1$

Case 3) $|b|$ = 1, which is in the range.

Case 5) $b^2 = -a$ , hence $|b| = 1$

Case 6) Since $a \neq 0$ , $(1+b) = 0$ , $|b| = 1$

Case 4) If $b = 0$ , $f(z) = \frac{z+a}{z}$ , $f(0)$ is not define.

In this case, $f(f(f(f(\frac{a}{a-1})))) = f(f(f(a)))= f(f(2)) = f( \frac{2+a}{2}) = \frac{2+3a}{2+a}$ and $f(f(f(f(\frac{a}{a-2})))) = f(f(f(2a)))= f(f(\frac{3}{2})) = f( \frac{3+2a}{3}) = \frac{3+5a}{3+2a}$

Hence, $(2+a)a = (a-1)(2+3a)$ and $(a)(3+2a) = (a-2)(3+5a)$ . Once, you work out this system, you will get no solution with $|a| = 1$ .

$\boxed{\textbf{(C)}\ 2\sqrt{6}}$

Solution 2

By algebraic manipulations, we obtain $h(z)=g(g(z))=f(f(f(f(z))))=\frac{Pz+Q}{Rz+S}$ where $P=(a+1)^2+a(b+1)^2$ , $Q=a(b+1)(b^2+2a+1)$ , $R=(b+1)(b^2+2a+1)$ , and $S=a(b+1)^2+(a+b^2)^2$ . In order for $h(z)=z$ , we must have $R=0$ , $Q=0$ , and $P=S$ . The first implies $b=-1$ or $b^2+2a+1=0$ , the second implies $a=0$ , $b=-1$ , or $b^2+2a+1=0$ , and the third implies $b=\pm1$ or $b^2+2a+1=0$ . Since $|a|=1\neq 0$ , in order to satisfy all 3 conditions we must have either $b=1$ or $b^2+2a+1=0$ . In the first case $|b|=1$ . $\[\]$ For the latter case note that $|b^2+1|=|-2a|=2$

so,

$2=|b^2+1|\leq |b^2|+1$ and hence, $1\leq|b|^2\Rightarrow1\leq |b|$ . On the other hand, $2=|b^2+1|\geq|b^2|-1$ so, $|b^2|\leq 3\Rightarrow0\leq |b|\leq \sqrt{3}$ . Thus $1\leq |b|\leq \sqrt{3}$ . Hence the maximum value for $|b|$ is $\sqrt{3}$ while the minimum is $1$ (which can be achieved for instance when $|a|=1,|b|=\sqrt{3}$ or $|a|=1,|b|=1$ respectively). Therefore the answer is $\boxed{\textbf{(C)}\ 2\sqrt{6}}$ .

@@ Line 63: / Line 63: @@
 By algebraic manipulations, we obtain
-<cmath>h(z)=g(g(z))=f(f(f(f(z))))=\frac{Pz+Q}{Rz+S}</cmath> where <math>P=(a+1)^2+a(b+1)^2</math>, <math>Q=a(b+1)(b^2+2a+1)</math>, <math>R=(b+1)(b^2+2a+1)</math>, and <math>S=a(b+1)^2+(a+b^2)^2</math>. In order for <math>h(z)=z</math>, we must have <math>R=0</math>, <math>Q=0</math>, and <math>P=S</math>. The first implies <math>b=-1</math> or <math>b^2+2a+1=0</math>, the second implies <math>a=0</math>, <math>b=-1</math>, or <math>b^2+2a+1=0</math>, and the third implies <math>b=\pm1</math> or <math>b^2+2a+1=0</math>. Since <math>|a|=1\neq 0</math>, in order to satisfy all 3 conditions we must have either <math>b=1</math> or <math>b^2+2a+1=0</math>. In the first case <math>|b|=1</math>. For the latter case note that <math>|b^2+1|=|-2a|=2</math> so that <math>2=|b^2+1|\leq |b^2|+1</math> and hence <math>1\leq|b|^2\Rightarrow1\leq |b|</math>. On the other hand <math>2=|b^2+1|\geq|b^2|-1</math> so that <math>|b^2|\leq 3\Rightarrow0\leq |b|\leq \sqrt{3}</math>. Thus <math>1\leq |b|\leq \sqrt{3}</math>. Hence in any case the maximum value for <math>|b|</math> is <math>\sqrt{3}</math> while the minimum is <math>1</math> (which can be achieved for instance when <math>|a|=1,|b|=\sqrt{3}</math> or <math>|a|=1,|b|=1</math> respectively). Hence the answer is <math>\boxed{\textbf{(C)}\ 2\sqrt{6}}</math>.
+<cmath>h(z)=g(g(z))=f(f(f(f(z))))=\frac{Pz+Q}{Rz+S}</cmath> where <math>P=(a+1)^2+a(b+1)^2</math>, <math>Q=a(b+1)(b^2+2a+1)</math>, <math>R=(b+1)(b^2+2a+1)</math>, and <math>S=a(b+1)^2+(a+b^2)^2</math>. In order for <math>h(z)=z</math>, we must have <math>R=0</math>, <math>Q=0</math>, and <math>P=S</math>. The first implies <math>b=-1</math> or <math>b^2+2a+1=0</math>, the second implies <math>a=0</math>, <math>b=-1</math>, or <math>b^2+2a+1=0</math>, and the third implies <math>b=\pm1</math> or <math>b^2+2a+1=0</math>. Since <math>|a|=1\neq 0</math>, in order to satisfy all 3 conditions we must have either <math>b=1</math> or <math>b^2+2a+1=0</math>. In the first case <math>|b|=1</math>. <cmath> </cmath>
+For the latter case note that
+<cmath>|b^2+1|=|-2a|=2</cmath>
+ so,
+<cmath>2=|b^2+1|\leq |b^2|+1</cmath>
+and hence,
+<cmath>1\leq|b|^2\Rightarrow1\leq |b|</cmath>.
+On the other hand,
+<cmath>2=|b^2+1|\geq|b^2|-1</cmath>
+so,
+<cmath>|b^2|\leq 3\Rightarrow0\leq |b|\leq \sqrt{3}</cmath>. Thus <math>1\leq |b|\leq \sqrt{3}</math>. Hence the maximum value for <math>|b|</math> is <math>\sqrt{3}</math> while the minimum is <math>1</math> (which can be achieved for instance when <math>|a|=1,|b|=\sqrt{3}</math> or <math>|a|=1,|b|=1</math> respectively). Therefore the answer is <math>\boxed{\textbf{(C)}\ 2\sqrt{6}}</math>.
 == See also ==

2011 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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