Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

m (Blanked the page)
(Tag: Blanking)
Line 1: Line 1:
 +
Let <math>P(x)</math> be the unique polynomial of minimal degree with the following properties:
  
 +
*<math>P(x)</math> has a leading coefficient <math>1</math>,
 +
 +
*<math>1</math> is a root of <math>P(x)-1</math>,
 +
 +
*<math>2</math> is a root of <math>P(x-2)</math>,
 +
 +
*<math>3</math> is a root of <math>P(3x)</math>, and
 +
 +
*<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>?

Revision as of 17:23, 9 November 2023

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 $m+n$?

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_AMC_10A_Problems/Problem_21&oldid=201090"