Difference between revisions of "2023 AMC 10A Problems/Problem 21"

Revision as of 07:55, 5 September 2024

Problem

Let $P(x)$ be the unique polynomial of minimal degree with the following properties:

$P(x)$ has a leading coefficient $1$ ,

$1$ is a root of $P(x)-1$ ,

$2$ is a root of $P(x-2)$ ,

$3$ is a root of $P(3x)$ , and

$4$ is a root of $4P(x)$ .

The roots of $P(x)$ are integers, with one exception. The root that is not an integer and can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime integers. What is $m+n$ ?

$\textbf{(A) }41\qquad\textbf{(B) }43\qquad\textbf{(C) }45\qquad\textbf{(D) }47\qquad\textbf{(E) }49$

Solution 1

From the problem statement, we know $P(2-2)=0$ , $P(9)=0$ and $4P(4)=0$ . Therefore, we know that $0$ , $9$ , and $4$ are roots. So, we can factor $P(x)$ as $x(x - 9)(x - 4)(x - a)$ , where $a$ is the unknown root. Since $P(x) - 1 = 0$ , we plug in $x = 1$ which gives $1(-8)(-3)(1 - a) = 1$ , therefore $24(1 - a) = 1 \implies 1 - a = 1/24 \implies a = 23/24$ . Therefore, our answer is $23 + 24 =\boxed{\textbf{(D) }47}$

~aiden22gao

~cosinesine

~walmartbrian

~sravya_m18

~ESAOPS

~Andrew_Lu

Solution 2

We proceed similarly to solution one. We get that $x(x-9)(x-4)(x-a)=1$ . Expanding, we get that $x(x-9)(x-4)(x-a)=x^4-(a+13)x^3+(13a+36)x^2-36ax$ . We know that $P(1)=1$ , so the sum of the coefficients of the cubic expression is equal to one. Thus $1-(a+13)+(13a+36)-36a=1$ . Solving for $a$ , we get that $a = 23/24$ . Therefore, our answer is $23 + 24 =\boxed{\textbf{(D) }47}$

~Aopsthedude

Video Solution 1 by Math-X (First fully understand the problem!!!)

https://youtu.be/GP-DYudh5qU?si=HDaV3kGwwywXP2J5&t=7475

~Math-X

Video Solution 2 by OmegaLearn

https://youtu.be/aOL04sKGyfU

Video Solution

https://youtu.be/jIqF_dhczsY

Video Solution by CosineMethod

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

Video Solution 3

https://youtu.be/Jan9feBsPEg

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

Video Solution 4 by EpicBird08

https://www.youtube.com/watch?v=D4GWjJmpqEU&t=25s

Video Solution 5 by MegaMath

https://www.youtube.com/watch?v=4Hwt3f1bi1c&t=1s

@@ Line 12: / Line 12: @@
 *<math>4</math> is a root of <math>4P(x)</math>.
-The roots of <math>P(x)</math> are integers, with one exception. The root that is not an integer can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime integers. What is <math>m+n</math>?
+The roots of <math>P(x)</math> are integers, with one exception. The root that is not an integer and can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime integers. What is <math>m+n</math>?
 <math>\textbf{(A) }41\qquad\textbf{(B) }43\qquad\textbf{(C) }45\qquad\textbf{(D) }47\qquad\textbf{(E) }49</math>
@@ Line 18: / Line 18: @@
 ==Solution 1==
-From the problem statement, we know <math>P(1)=1</math>, <math>P(2-2)=0</math>, <math>P(9)=0</math> and <math>4P(4)=0</math>. Therefore, we know that <math>0</math>, <math>9</math>, and <math>4</math> are roots. Because of this, we can factor <math>P(x)</math> as <math>x(x - 9)(x - 4)(x - a)</math>, where <math>a</math> is the unknown root. Plugging in <math>x = 1</math> gives <math>1(-8)(-3)(1 - a) = 1</math>, so <math>24(1 - a) = 1/24 \implies 1 - a = 24 \implies a = 23/24</math>. Therefore, our answer is <math>23 + 24 =</math> <math>47</math>, or <math>C</math>
+From the problem statement, we know <math>P(2-2)=0</math>, <math>P(9)=0</math> and <math>4P(4)=0</math>. Therefore, we know that <math>0</math>, <math>9</math>, and <math>4</math> are roots. So, we can factor <math>P(x)</math> as <math>x(x - 9)(x - 4)(x - a)</math>, where <math>a</math> is the unknown root. Since <math>P(x) - 1 = 0</math>, we plug in <math>x = 1</math> which gives <math>1(-8)(-3)(1 - a) = 1</math>, therefore <math>24(1 - a) = 1 \implies 1 - a = 1/24 \implies a = 23/24</math>. Therefore, our answer is <math>23 + 24 =\boxed{\textbf{(D) }47}</math>
 ~aiden22gao
@@ Line 26: / Line 26: @@
 ~walmartbrian
+~sravya_m18
-== Video Solution 1 by OmegaLearn ==
+~ESAOPS
+~Andrew_Lu
+== Solution 2==
+We proceed similarly to solution one. We get that <math>x(x-9)(x-4)(x-a)=1</math>. Expanding, we get that <math>x(x-9)(x-4)(x-a)=x^4-(a+13)x^3+(13a+36)x^2-36ax</math>. We know that <math>P(1)=1</math>, so the sum of the coefficients of the cubic expression is equal to one. Thus <math>1-(a+13)+(13a+36)-36a=1</math>. Solving for <math>a</math>, we get that <math>a = 23/24</math>. Therefore, our answer is <math>23 + 24 =\boxed{\textbf{(D) }47}</math>
+~Aopsthedude
+==Video Solution 1 by Math-X (First fully understand the problem!!!)==
+https://youtu.be/GP-DYudh5qU?si=HDaV3kGwwywXP2J5&t=7475
+~Math-X
+== Video Solution 2 by OmegaLearn ==
 https://youtu.be/aOL04sKGyfU
+== Video Solution ==
+https://youtu.be/jIqF_dhczsY
+== Video Solution by CosineMethod ==
+https://www.youtube.com/watch?v=HEqewKGKrFE
+==Video Solution 3==
+https://youtu.be/Jan9feBsPEg
+~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
+==Video Solution 4 by EpicBird08==
+https://www.youtube.com/watch?v=D4GWjJmpqEU&t=25s
+==Video Solution 5 by MegaMath==
+https://www.youtube.com/watch?v=4Hwt3f1bi1c&t=1s
 ==See Also==
 {{AMC10 box|year=2023|ab=A|num-b=20|num-a=22}}
 {{MAA Notice}}

2023 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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 10 Problems and Solutions

Page

Toolbox

Search