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Difference between revisions of "2024 AMC 12A Problems/Problem 25"

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<math>\textbf{(A) }1282\qquad\textbf{(B) }1292\qquad\textbf{(C) }1310\qquad\textbf{(D) }1320\qquad\textbf{(E) }1330</math>
 
<math>\textbf{(A) }1282\qquad\textbf{(B) }1292\qquad\textbf{(C) }1310\qquad\textbf{(D) }1320\qquad\textbf{(E) }1330</math>
−
==Solution 1==
+
==Solution 1 (Inverse Function)==
−
Symmetric about the line <math>y=x</math> implies that the inverse fuction <math>y^{-1}=y</math>. Then we split the question into several cases to find the final answer.
+
Symmetric about the line <math>y=x</math> implies that the inverse function <math>y^{-1}=y</math>. Then we split the question into several cases to find the final answer.
  
  
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And <math>y=\frac{(cx+d)(\frac{a}{c})+b-\frac{ad}{c}}{cx+d}=\frac{a}{c}+\frac{b-\frac{ad}{c}}{cx+d}</math>
 
And <math>y=\frac{(cx+d)(\frac{a}{c})+b-\frac{ad}{c}}{cx+d}=\frac{a}{c}+\frac{b-\frac{ad}{c}}{cx+d}</math>
  
−
So <math>\frac{a}{c}=-\frac{d}{c}</math>, or <math>a=-d</math> (<math>c\neq 0</math>), and substitude that into <math>\frac{b-\frac{ad}{c}}{cx-a}=\frac{b-\frac{ad}{c}}{cx+d}</math> gives us:
+
So <math>\frac{a}{c}=-\frac{d}{c}</math>, or <math>a=-d</math> (<math>c\neq 0</math>), and substitute that into <math>\frac{b-\frac{ad}{c}}{cx-a}=\frac{b-\frac{ad}{c}}{cx+d}</math> gives us:
  
 
<math>bc-ad\neq 0</math> (Otherwise <math>y=\frac{a}{c}</math>, <math>y^{-1}=-\frac{d}{c}=\frac{a}{c}</math>, and is not symmetric about <math>y=x</math>)
 
<math>bc-ad\neq 0</math> (Otherwise <math>y=\frac{a}{c}</math>, <math>y^{-1}=-\frac{d}{c}=\frac{a}{c}</math>, and is not symmetric about <math>y=x</math>)
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<math>a=0</math>: <math>10\times 10=100</math> ways.
 
<math>a=0</math>: <math>10\times 10=100</math> ways.
  
−
<math>a=-1,1</math>: <math>(11\times 10-2)\times 2=216</math> ways.
+
<math>a=\pm 1</math>: <math>(11\times 10-2)\times 2=216</math> ways.
  
−
<math>a=-2,2</math>: <math>(11\times 10-2)\times 2=216</math> ways.
+
<math>a=\pm 2</math>: <math>(11\times 10-2)\times 2=216</math> ways.
  
−
<math>a=-3,3</math>: <math>(11\times 10-2)\times 2=216</math> ways.
+
<math>a=\pm 3</math>: <math>(11\times 10-2)\times 2=216</math> ways.
  
−
<math>a=-4,4</math>: <math>(11\times 10-6)\times 2=208</math> ways.
+
<math>a=\pm 4</math>: <math>(11\times 10-6)\times 2=208</math> ways.
  
−
<math>a=-5,5</math>: <math>(11\times 10-2)\times 2=216</math> ways.
+
<math>a=\pm 5</math>: <math>(11\times 10-2)\times 2=216</math> ways.
  
 
In total <math>100+208+216\times 4= 1172</math> ways.
 
In total <math>100+208+216\times 4= 1172</math> ways.
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==Solution 2 (Rotation + Edge Cases)==
 
==Solution 2 (Rotation + Edge Cases)==
−
First, observe that the only linear functions that are symmetric about <math>y = x</math> are <math>y = x</math> and <math>y = -x</math>.  
+
First, observe that the only linear functions that are symmetric about <math>y = x</math> are <math>y = x</math> and <math>y = -x+k</math>, where <math>k</math> is some constant.  
  
  
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<cmath>\frac{v + u}{\sqrt{2}} = \frac{a(v - u) + b\sqrt{2}}{c(v - u) + d\sqrt{2}}</cmath>
 
<cmath>\frac{v + u}{\sqrt{2}} = \frac{a(v - u) + b\sqrt{2}}{c(v - u) + d\sqrt{2}}</cmath>
  
−
Multiplying both equations by <math>\sqrt{2}</math> and subtracting the second equation from the first equation gives <math>d = -a</math>. Since <math>a, b, c, d</math> are integers between <math>-5</math> and <math>5</math>, this gives <math>11^3 = 1331</math> combinations. We need to subtract the edge cases that don't work, namely all undefined functions and linear functions except <math>y = x</math> and <math>y = -x</math>. Consider the following cases:
+
Multiplying both equations by <math>\sqrt{2}</math> and subtracting the second equation from the first equation gives <math>d = -a</math>. Since <math>a, b, c, d</math> are integers between <math>-5</math> and <math>5</math>, this gives <math>11^3 = 1331</math> combinations. We need to subtract the edge cases that don't work, namely all undefined functions and linear functions except <math>y = x</math> and <math>y = -x+k</math>. Consider the following cases:
  
  
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−
Finally, we need to add the case where <math>y = x</math>, which occurs when <math>a = d</math> and <math>b = c = 0</math>. <math>a</math> can be any integer from <math>-5</math> to <math>5</math> except <math>0</math>, so this case has <math>10</math> combinations. Since <math>y = -x</math> occurs when <math>a = -d</math> and <math>b = c = 0</math>, this case is already counted.
+
Finally, we need to add the case where <math>y = x</math>, which occurs when <math>a = d</math> and <math>b = c = 0</math>. <math>a</math> can be any integer from <math>-5</math> to <math>5</math> except <math>0</math>, so this case has <math>10</math> combinations. Since <math>y = -x+k</math> occurs when <math>a = -d</math> and <math>c = 0</math>, this case has already been counted.
  
  
  
−
The answer is <math>1331 - 28 - 21 + 10 = \boxed{\textbf{(B) }1292}</math>.
+
Thus, the answer is <math>1331 - 28 - 21 + 10 = \boxed{\textbf{(B) }1292}</math>.
  
 
~babyhamster
 
~babyhamster
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== Solution 3 (Asymptotes) ==
 
== Solution 3 (Asymptotes) ==
 
There are two cases: when <math>c=0</math> and when <math>c\neq 0</math>.
 
There are two cases: when <math>c=0</math> and when <math>c\neq 0</math>.
 +
 +
  
 
<math>\textbf{Case 1: }\mathbf{c=0.}</math> If <math>c=0</math>, then <math>y=\frac{ax+b}d=\frac adx+\frac bd</math>. This is the equation of a line, and the only lines symmetric about <math>y=x</math> are those perpendicular to <math>y=x</math> (i.e. those with slope <math>-1</math>) and <math>y=x</math> itself. To have a slope of <math>-1</math>, we need <math>a=-d\neq 0</math>, and <math>b</math> can be any of its <math>11</math> possibilities from <math>-5</math> to <math>5</math>. There are <math>11\cdot 10=110</math> possibilities here. For the function to be <math>y=x</math>, we need <math>a=d\neq 0</math> and <math>b=0</math>. There are <math>10</math> possibilities here. Thus, our total for Case 1 is <math>110+10=120</math> possiblities.
 
<math>\textbf{Case 1: }\mathbf{c=0.}</math> If <math>c=0</math>, then <math>y=\frac{ax+b}d=\frac adx+\frac bd</math>. This is the equation of a line, and the only lines symmetric about <math>y=x</math> are those perpendicular to <math>y=x</math> (i.e. those with slope <math>-1</math>) and <math>y=x</math> itself. To have a slope of <math>-1</math>, we need <math>a=-d\neq 0</math>, and <math>b</math> can be any of its <math>11</math> possibilities from <math>-5</math> to <math>5</math>. There are <math>11\cdot 10=110</math> possibilities here. For the function to be <math>y=x</math>, we need <math>a=d\neq 0</math> and <math>b=0</math>. There are <math>10</math> possibilities here. Thus, our total for Case 1 is <math>110+10=120</math> possiblities.
 +
 +
  
 
<math>\textbf{Case 2: }\mathbf{c\neq 0.}</math> When <math>c\neq 0</math>, we will first consider the case in which the graph is a hyperbola. Clearly, for this hyperbola to be symmetric about <math>y=x</math>, the intersection of its horizontal and vertical asymptotes must be on <math>y=x</math>. The location of the horizontal asymptote is <math>y=\frac ac</math>, and the vertical asymptote occurs at <math>x=-\frac dc</math>. These asymptotes intersect on <math>y=x</math> when <math>\frac ac = -\frac dc</math>, or, more simply, when <math>a=-d</math>.
 
<math>\textbf{Case 2: }\mathbf{c\neq 0.}</math> When <math>c\neq 0</math>, we will first consider the case in which the graph is a hyperbola. Clearly, for this hyperbola to be symmetric about <math>y=x</math>, the intersection of its horizontal and vertical asymptotes must be on <math>y=x</math>. The location of the horizontal asymptote is <math>y=\frac ac</math>, and the vertical asymptote occurs at <math>x=-\frac dc</math>. These asymptotes intersect on <math>y=x</math> when <math>\frac ac = -\frac dc</math>, or, more simply, when <math>a=-d</math>.
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Adding the possibilities for Case 1 and Case 2 yields our final answer of <math>120+1172=\boxed{\textbf{(B) }1292}</math> possible quadruples.
 
Adding the possibilities for Case 1 and Case 2 yields our final answer of <math>120+1172=\boxed{\textbf{(B) }1292}</math> possible quadruples.
 +
 +
== Solution 4 ==
 +
Note that the condition is equivalent to having <math>f(f(x))=x</math>.
 +
 +
So we have: <math>\frac{a(\frac{ax+b}{cx+d})+b}{c(\frac{ax+b}{cx+d})+d} = x \Rightarrow x^2(ac+cd)+x(d^2-a^2)-(ab+bd)=0</math>
 +
 +
Thus we require:
 +
 +
<math>ac+cd = 0 \rightarrow c=0</math> or <math>a = -d</math> 
 +
 +
<math>d^2-a^2 = 0 \rightarrow a = d</math> or <math>a = -d</math>
 +
 +
<math>ab+bd = 0 \rightarrow b=0</math> or <math>a = -d</math>
 +
 +
Note that if <math>a = -d</math> then all 3 cases work and give <math>11^3=1331</math> solutions.
 +
 +
If instead <math>c=0</math> and <math>a \neq -d</math> then we require <math>a=d</math> and <math>b=0</math> which then give <math>10</math> solutions.
 +
 +
Now, we must remove all extraneous cases. This is when <math>x(ac+cd)+bc+d^2 = 0</math> (note this includes the case where <math>c=d=0</math>).
 +
 +
So this is equivalent to having both <math>ac+cd = 0</math> and <math>bc + d^2 = 0.</math>
 +
 +
If <math>c = d = 0</math> we have <math>11</math> solutions.
 +
 +
If <math>c = 0</math> and <math>d \neq 0</math> then we have <math>0</math> solutions.
 +
 +
If <math>c \neq 0</math> and <math>d = 0</math> then we require <math>a=0</math> and <math>b=0</math> so we have <math>10</math> solutions.
 +
 +
And if <math>c \neq 0</math> and <math>d \neq 0</math> we have <math>a = -d</math> and <math>d^2 = -bc</math>, see that if <math>\left|d\right|=1,3,4,5</math> we have <math>2</math> solutions and if <math>\left|d\right|=2</math> we have <math>6</math> solutions, so a total of <math>2(2+6+2+2+2) = 28</math> solutions.
 +
 +
Thus, the final answer is <math>1331+10-11-10-28 =\boxed{\textbf{(B) }1292}</math>
 +
 +
~LuisFonseca123
 +
  
 
==See also==
 
==See also==
 
{{AMC12 box|year=2024|ab=A|num-b=24|after=Last Problem}}
 
{{AMC12 box|year=2024|ab=A|num-b=24|after=Last Problem}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 12:28, 10 November 2024

Problem

A graph is $\textit{symmetric}$ about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers $(a,b,c,d)$, where $|a|,|b|,|c|,|d|\le5$ and $c$ and $d$ are not both $0$, is the graph of \[y=\frac{ax+b}{cx+d}\]symmetric about the line $y=x$?

$\textbf{(A) }1282\qquad\textbf{(B) }1292\qquad\textbf{(C) }1310\qquad\textbf{(D) }1320\qquad\textbf{(E) }1330$

Solution 1 (Inverse Function)

Symmetric about the line $y=x$ implies that the inverse function $y^{-1}=y$. Then we split the question into several cases to find the final answer.


Case 1: $c=0$

Then $y=\frac{a}{d}x+\frac{b}{d}$ and $y^{-1}=\frac{d}{a}x-\frac{b}{a}$. Giving us $\frac{a}{d}=\frac{d}{a}$ and $\frac{b}{d}=-\frac{b}{a}$

Therefore, we obtain 2 subcases: $b\neq 0, a+d=0$ and $b=0, a^2=d^2$


Case 2: $c\neq 0$

Then $y^{-1}=\frac{b-dx}{cx-a}=\frac{(cx-a)(-\frac{d}{c})+b-\frac{ad}{c}}{cx-a}=-\frac{d}{c}+\frac{b-\frac{ad}{c}}{cx-a}$

And $y=\frac{(cx+d)(\frac{a}{c})+b-\frac{ad}{c}}{cx+d}=\frac{a}{c}+\frac{b-\frac{ad}{c}}{cx+d}$

So $\frac{a}{c}=-\frac{d}{c}$, or $a=-d$ ($c\neq 0$), and substitute that into $\frac{b-\frac{ad}{c}}{cx-a}=\frac{b-\frac{ad}{c}}{cx+d}$ gives us:

$bc-ad\neq 0$ (Otherwise $y=\frac{a}{c}$, $y^{-1}=-\frac{d}{c}=\frac{a}{c}$, and is not symmetric about $y=x$)


Therefore we get three cases:

Case 1.1: $c= 0, b\neq 0, d\neq 0, a+d=0$

We have 10 choice of $b$, 10 choice of $d$ and each choice of $d$ has one corresponding choice of $a$. In total $10\times 10=100$ ways.


Case 1.2: $c= 0, b = 0, d\neq 0, a^2=d^2$

We have 10 choice for $d$ ($d\neq 0$), each choice of $d$ has 2 corresponding choice of $a$, thus $10\times 2=20$ ways.


Case 2: $c\neq 0, bc-ad\neq 0, a=-d$

$a=0$: $10\times 10=100$ ways.

$a=\pm 1$: $(11\times 10-2)\times 2=216$ ways.

$a=\pm 2$: $(11\times 10-2)\times 2=216$ ways.

$a=\pm 3$: $(11\times 10-2)\times 2=216$ ways.

$a=\pm 4$: $(11\times 10-6)\times 2=208$ ways.

$a=\pm 5$: $(11\times 10-2)\times 2=216$ ways.

In total $100+208+216\times 4= 1172$ ways.


So the answer is $100+20+1172= \boxed{\textbf{(B) }1292}$

~ERiccc

Solution 2 (Rotation + Edge Cases)

First, observe that the only linear functions that are symmetric about $y = x$ are $y = x$ and $y = -x+k$, where $k$ is some constant.


We perform a $45^\circ$ counterclockwise rotation of the Cartesian plane. Let $(x, y)$ be sent to $(u, v)$. Then $u$ and $v$ are the real and imaginary parts of $(x + yi)(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)$ respectively, which gives

\[u = \frac{x - y}{\sqrt{2}}\] \[v = \frac{x + y}{\sqrt{2}}\]

so

\[x = \frac{v + u}{\sqrt{2}}\] \[y = \frac{v - u}{\sqrt{2}}\].

The rotated function is symmetric about the y-axis, so the equation holds after replacing all instances of $u$ with $-u$ (this is just switching the values of $x$ and $y$ which is a reflection over $y = x$, but working in terms of $(u, v)$ allows more cancellations in the following calculations).

Writing $x$ and $y$ in terms of $u$ and $v$, we have

\[\frac{v - u}{\sqrt{2}} = \frac{a(v + u) + b\sqrt{2}}{c(v + u) + d\sqrt{2}}\] \[\frac{v + u}{\sqrt{2}} = \frac{a(v - u) + b\sqrt{2}}{c(v - u) + d\sqrt{2}}\]

Multiplying both equations by $\sqrt{2}$ and subtracting the second equation from the first equation gives $d = -a$. Since $a, b, c, d$ are integers between $-5$ and $5$, this gives $11^3 = 1331$ combinations. We need to subtract the edge cases that don't work, namely all undefined functions and linear functions except $y = x$ and $y = -x+k$. Consider the following cases:


Case 1: $a, b, c, d$ are all nonzero. Then the function is linear when $ax + b$ is a multiple of $cx + d$, or $\frac{a}{b} = \frac{c}{-a}$.

If $a = \pm 1$, $(b,c) = (1, -1)$ or $(-1, 1)$; there are $2*2 = 4$ ways.

If $a = \pm 2$, there are $12$ ways.

If $a = \pm 3$, there are $4$ ways.

If $a = \pm 4$, there are $4$ ways.

If $a = \pm 5$, there are $4$ ways.

In total, this case has $28$ combinations.


Case 2: $a = b = d = 0$ or $a = c = d = 0$

If $a = b = d = 0$ then $c$ can take on $11$ values, and if $a = c = d = 0$, then $b$ can take on $11$ values, but $a = b = c = d = 0$ is counted twice so this case has $11 + 11 - 1 = 21$ combinations.


Finally, we need to add the case where $y = x$, which occurs when $a = d$ and $b = c = 0$. $a$ can be any integer from $-5$ to $5$ except $0$, so this case has $10$ combinations. Since $y = -x+k$ occurs when $a = -d$ and $c = 0$, this case has already been counted.


Thus, the answer is $1331 - 28 - 21 + 10 = \boxed{\textbf{(B) }1292}$.

~babyhamster

Solution 3 (Asymptotes)

There are two cases: when $c=0$ and when $c\neq 0$.


$\textbf{Case 1: }\mathbf{c=0.}$ If $c=0$, then $y=\frac{ax+b}d=\frac adx+\frac bd$. This is the equation of a line, and the only lines symmetric about $y=x$ are those perpendicular to $y=x$ (i.e. those with slope $-1$) and $y=x$ itself. To have a slope of $-1$, we need $a=-d\neq 0$, and $b$ can be any of its $11$ possibilities from $-5$ to $5$. There are $11\cdot 10=110$ possibilities here. For the function to be $y=x$, we need $a=d\neq 0$ and $b=0$. There are $10$ possibilities here. Thus, our total for Case 1 is $110+10=120$ possiblities.


$\textbf{Case 2: }\mathbf{c\neq 0.}$ When $c\neq 0$, we will first consider the case in which the graph is a hyperbola. Clearly, for this hyperbola to be symmetric about $y=x$, the intersection of its horizontal and vertical asymptotes must be on $y=x$. The location of the horizontal asymptote is $y=\frac ac$, and the vertical asymptote occurs at $x=-\frac dc$. These asymptotes intersect on $y=x$ when $\frac ac = -\frac dc$, or, more simply, when $a=-d$.

If the asymptotes intersect on $y=x$, then the hyperbola must be symmetric about $y=x$. This is true because for any hyperbola with perpendicular asymptotes, we can rotate and translate the coordinate plane in a certain way such that that hyperbola has an equation of the form $x^2-y^2=a^2$. Then, the hyperbola's asymptotes would intersect at the origin, and it would be symmetric about the coordinate axes (because it makes a distinction neither between $x$ and $-x$ nor $y$ and $-y$). The coordinate axes are the bisectors of the angles formed by the asymptotes, and the hyperbola is symmetric about them. Thus, because the angles formed by our hyperbola's asymptotes are bisected by $y=x$, our hyperbola must be symmetric about $y=x$.

Thus, with the conditions that $a=-d$ and $c\neq 0$, there are $11\cdot 11\cdot 10\cdot 1=1210$ possibilites for $(a,b,c,d)$. However, not all of these ordered quadruples produce hyperbolas. If $b=a=d=0$ or $\frac ac=\frac bd$, then the quadruples produce horizontal lines with a hole when the denominator equals $0$. As seen in Case 1, these lines, with slope $0$, cannot be symmetric about $y=x$.

For the subcase where $b=a=d=0$, there are $10$ possibilities for $c\neq 0$, which gives us $10$ wrongly counted quadruples.

For the subcase where $\frac ac=\frac bd$, we wrongly counted cases where $d=-a$. Here, $-a^2=bc$ by cross-multiplication. The casework on the possible values of $a$ below counts the number of triples $(a,b,c)$ with $|a|,|b|,|c|\leq 5$ which satisfy this condition.

If $a=\pm 1$, $bc=-1$, which yields $2\cdot 2 \cdot 1=4$ possibilities.

If $a=\pm 2$, $bc=-4$, which yields $2\cdot 2\cdot 3=12$ possibilities.

If $a=\pm 3$, $bc=-9$, which yields $2\cdot 2\cdot 1=4$ possbilities. (recall that $|b|,|c|\leq 5$)

If $a=\pm 4$, $bc=-16$, which yields $2\cdot 2\cdot 1=4$ possibilities.

If $a=\pm 5$, $bc=-25$, which yields $2\cdot 2\cdot 1=4$ possibilities.

Adding the above values together for this subcase yields $4+12+4+4+4=28$ wrongly counted quadruples.

Subtracting the wrongly counted quadruples from our count for Case 2 yields $1210-10-28=1172$.

Adding the possibilities for Case 1 and Case 2 yields our final answer of $120+1172=\boxed{\textbf{(B) }1292}$ possible quadruples.

Solution 4

Note that the condition is equivalent to having $f(f(x))=x$.

So we have: $\frac{a(\frac{ax+b}{cx+d})+b}{c(\frac{ax+b}{cx+d})+d} = x \Rightarrow x^2(ac+cd)+x(d^2-a^2)-(ab+bd)=0$

Thus we require:

$ac+cd = 0 \rightarrow c=0$ or $a = -d$

$d^2-a^2 = 0 \rightarrow a = d$ or $a = -d$

$ab+bd = 0 \rightarrow b=0$ or $a = -d$

Note that if $a = -d$ then all 3 cases work and give $11^3=1331$ solutions.

If instead $c=0$ and $a \neq -d$ then we require $a=d$ and $b=0$ which then give $10$ solutions.

Now, we must remove all extraneous cases. This is when $x(ac+cd)+bc+d^2 = 0$ (note this includes the case where $c=d=0$).

So this is equivalent to having both $ac+cd = 0$ and $bc + d^2 = 0.$

If $c = d = 0$ we have $11$ solutions.

If $c = 0$ and $d \neq 0$ then we have $0$ solutions.

If $c \neq 0$ and $d = 0$ then we require $a=0$ and $b=0$ so we have $10$ solutions.

And if $c \neq 0$ and $d \neq 0$ we have $a = -d$ and $d^2 = -bc$, see that if $\left|d\right|=1,3,4,5$ we have $2$ solutions and if $\left|d\right|=2$ we have $6$ solutions, so a total of $2(2+6+2+2+2) = 28$ solutions.

Thus, the final answer is $1331+10-11-10-28 =\boxed{\textbf{(B) }1292}$

~LuisFonseca123


See also

2024 AMC 12A (Problems • Answer Key • Resources)
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