Difference between revisions of "2023 AMC 10B Problems/Problem 22"

Revision as of 10:10, 19 November 2023

Problem

How many distinct values of 𝑥 satisfy $\lfloor{x}\rfloor^2-3x+2=0$ , where $\lfloor{x}\rfloor$ denotes the largest integer less than or equal to $x$ ?

$\textbf{(A) } \text{an infinite number} \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 0$

Solution 1(three cases)

First, let's take care of the integer case--clearly, only $x=1,2$ work. Then, we know that $3x$ must be an integer. Set $x=\frac{a}3$ . Now, there are two cases for the value of $\lfloor x\rfloor$ . Case 1: $\lfloor x\rfloor=\frac{a-1}{3}$ $\frac{a^2-2a+1}{9}=a-2\rightarrow a^2-2a+1=9a-18\rightarrow a^2-11a+19=0.$ There are no solutions in this case. Case 2: $\lfloor x\rfloor=\frac{a-2}{3}$ $\frac{a^2-4a+4}{9}=a-2\rightarrow a^2-4a+4=9a-18\rightarrow a^2-13a+22=0.$ This case provides the two solutions $\frac23$ and $\frac{11}3$ as two more solutions. Our final answer is thus $\boxed{4}$ .

~wuwang2002

Solution 2

First, $x=2,1$ are trivial solutions

We assume from the shape of a parabola and the nature of the floor function that any additional roots will be near 2 and 1

We can now test values for $\lfloor{x}\rfloor$ :

$\lfloor{x}\rfloor=0$

We have $0-3x+2=0$ . Solving, we have $x=\frac{2}{3}$ . We see that $\lfloor{\frac{2}{3}}\rfloor=0$ , so this solution is valid

$\lfloor{x}\rfloor=-1$

We have $1-3x+2=0$ . Solving, we have $x=1$ . $\lfloor{1}\rfloor\neq-1$ , so this is not valid. We assume there are no more solutions in the negative direction and move on to $\lfloor{x}\rfloor=3$

$\lfloor{x}\rfloor=3$

We have $9-3x+2=0$ . Solving, we have $x=\frac{11}{3}$ . We see that $\lfloor{\frac{11}{3}}\rfloor=3$ , so this solution is valid

$\lfloor{x}\rfloor=4$

We have $16-3x+2=0$ . Solving, we have $x=6$ . $\lfloor{6}\rfloor\neq4$ , so this is not valid. We assume there are no more solutions.

Our final answer is $\boxed{\textbf{(B) }4}$

~kjljixx

Solution 3

Denote $a = \lfloor x \rfloor$ . Denote $b = x - \lfloor x \rfloor$ . Thus, $b \in \left[ 0 , 1 \right)$ .

The equation given in this problem can be written as $a^2 - 3 \left( a + b \right) + 2 = 0 .$

Thus, $\begin{align*} 3 b & = a^2 - 3 a + 2 . \end{align*}$

Because $b \in \left[ 0 , 1 \right)$ , we have $3 b \in \left[ 0 , 3 \right)$ . Thus, $a^2 - 3 a + 2 = 0, 1, \mbox{ or } 2 .$

If $a^2-3a+2=0$ , $(a-2)(a-1)=0$ so $a$ can be $1, 2$ .

If $a^2-3a+2=1$ , $a^2-3a+1=0$ which we find has no integer solutions after finding the discriminant.

If $a^2-3a+2=2$ , $a^2-3a=0$ -> $a(a-3)=0$ so $a$ can also be $0, 3$ .

Therefore, $a = 1$ , 2, 0, 3. Therefore, the number of solutions is $\boxed{\textbf{(B) 4}}$ .

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Solution 4(Quick)

A quadratic equation can have up to 2 real solutions. With the $\lfloor{x}\rfloor$ , it could also help generate another pair. We have to verify that the solutions are real and distinct.

First, we get the trivial solution by ignoring the floor. $(x-2)(x-1) = 0$ , we get $(2,1)$ as our first pair of solutions.

Up to this point, we can rule out A,E.

Next, we see that $\lfloor{x}\rfloor^2-3x=0.$ This implies that $-3x$ must be an integer. We can guess and check $x$ as $\dfrac{k}{3}$ which yields $\left(\dfrac{2}{3},\dfrac{11}{3}\right).$

So we got 4 in total $\left(\dfrac{2}{3},1,2,\dfrac{11}{3}\right).$

~Technodoggo

Solution 5

$x=1, 2$ are trivial solutions. Let $x=n+f$ for some integer $n$ and some number $f$ such that $-1<f<1$ . $\lfloor{x}\rfloor^2-3x+2= \lfloor{n+f}\rfloor^2-3(n+f)+2=n^2+-3(n+f)+2.$ So now we have $n^2-3(n+f)+2 = 0,$ which we can rewrite as $n(n-3)=3f-2.$ Since $n$ is an integer, $n(n-3)$ is an integer, so $3f-2$ is an integer. Since $-1<f<1$ , the only possible values of $f$ are $\frac{1}{3}$ , $\frac{2}{3}$ , $-\frac{1}{3}$ , and $-\frac{2}{3}$ . Plugging in each value, we find that the only value of $f$ that produces integer solutions for $n$ is $f=\frac{2}{3}$ . If $f=\frac{2}{3}$ , $n=0$ or $n=3$ . Hence, there is a total of 4 possible solutions, so the answer is $\boxed{\textbf{(B) }4}$ . ~azc1027

Video Solution 1 by OmegaLearn

https://youtu.be/wAYcpn-Q_KQ

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=DvHGEXBjf0Y

@@ Line 108: / Line 108: @@
 <cmath>n^2-3(n+f)+2 = 0,</cmath>
 which we can rewrite as
-<cmath>n(n+3)=3f-2.</cmath>
+<cmath>n(n-3)=3f-2.</cmath>
-Since <math>n</math> is an integer, <math>n(n+3)</math> is an integer, so <math>3f-2</math> is an integer. Since <math>-1<f<1</math>, the only possible values of <math>f</math> are <math>\frac{1}{3}</math>, <math>\frac{2}{3}</math>, <math>-\frac{1}{3}</math>, and <math>-\frac{2}{3}</math>. Plugging in each value, we find that the only value of <math>f</math> that produces integer solutions for <math>n</math> is <math>f=\frac{2}{3}</math>. If <math>f=\frac{2}{3}</math>, <math>n=0</math> or <math>n=3</math>. Hence, there is a total of 4 possible solutions, so the answer is <math>\boxed{\textbf{(B) }4}</math>.
+Since <math>n</math> is an integer, <math>n(n-3)</math> is an integer, so <math>3f-2</math> is an integer. Since <math>-1<f<1</math>, the only possible values of <math>f</math> are <math>\frac{1}{3}</math>, <math>\frac{2}{3}</math>, <math>-\frac{1}{3}</math>, and <math>-\frac{2}{3}</math>. Plugging in each value, we find that the only value of <math>f</math> that produces integer solutions for <math>n</math> is <math>f=\frac{2}{3}</math>. If <math>f=\frac{2}{3}</math>, <math>n=0</math> or <math>n=3</math>. Hence, there is a total of 4 possible solutions, so the answer is <math>\boxed{\textbf{(B) }4}</math>.
 ~azc1027

2023 AMC 10B (Problems • Answer Key • Resources)
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