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

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==Solution 1==
 
==Solution 1==
  
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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(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 =\boxed{\textbf{(D) }47}</math>.
  
 
~aiden22gao  
 
~aiden22gao  

Revision as of 20:46, 9 November 2023

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 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 $P(1)=1$, $P(2-2)=0$, $P(9)=0$ and $4P(4)=0$. Therefore, we know that $0$, $9$, and $4$ are roots. Because of this, we can factor $P(x)$ as $x(x - 9)(x - 4)(x - a)$, where $a$ is the unknown root. Plugging in $x = 1$ gives $1(-8)(-3)(1 - a) = 1$, so $24(1 - a) = 1/24 \implies 1 - a = 24 \implies a = 23/24$. Therefore, our answer is $23 + 24 =\boxed{\textbf{(D) }47}$.

~aiden22gao

~cosinesine

~walmartbrian


Video Solution 1 by OmegaLearn

https://youtu.be/aOL04sKGyfU

See Also

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

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