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 IMO Problems/Problem 4"

m (→‎Problem 4)
(→‎Solution)
Line 5: Line 5:
  
 
==Solution==
 
==Solution==
 +
We first solve for <math>a_1.</math> <math>a_1 = \sqrt{(x_1)(\frac{1}{x_1})} = \sqrt{1} = 1.</math>
 +
Now we solve for <math>a_{n+1}</math> in terms of <math>a_n</math> and <math>x.</math>

Revision as of 23:47, 13 July 2023

Problem

Let $x_1, x_2, \cdots , x_{2023}$ be pairwise different positive real numbers such that \[a_n = \sqrt{(x_1+x_2+···+x_n)(\frac1{x_1} + \frac1{x_2} +···+\frac1{x_n})}\] is an integer for every $n = 1,2,\cdots,2023$. Prove that $a_{2023} \ge 3034$.

Solution

We first solve for $a_1.$ $a_1 = \sqrt{(x_1)(\frac{1}{x_1})} = \sqrt{1} = 1.$ Now we solve for $a_{n+1}$ in terms of $a_n$ and $x.$

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_IMO_Problems/Problem_4&oldid=195693"