Difference between revisions of "2023 IMO Problems/Problem 4"
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Let <math>x_1, x_2, \cdots , x_{2023}</math> be pairwise different positive real numbers such that | Let <math>x_1, x_2, \cdots , x_{2023}</math> be pairwise different positive real numbers such that | ||
<cmath>a_n = \sqrt{(x_1+x_2+···+x_n)(\frac1{x_1} + \frac1{x_2} +···+\frac1{x_n})}</cmath> | <cmath>a_n = \sqrt{(x_1+x_2+···+x_n)(\frac1{x_1} + \frac1{x_2} +···+\frac1{x_n})}</cmath> | ||
is an integer for every <math>n = 1,2,\cdots,2023</math>. Prove that <math>a_{2023} \ge 3034</math>. | is an integer for every <math>n = 1,2,\cdots,2023</math>. Prove that <math>a_{2023} \ge 3034</math>. | ||
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| + | ==Solution== | ||
Revision as of 22:37, 13 July 2023
Problem
Let 
be pairwise different positive real numbers such that
![\[a_n = \sqrt{(x_1+x_2+···+x_n)(\frac1{x_1} + \frac1{x_2} +···+\frac1{x_n})}\]](http://latex.artofproblemsolving.com/9/e/d/9edc277a4c56ac586454738c4b2238a040695b47.png)
is an integer for every 
. Prove that 
.
