Difference between revisions of "2021 IMO Problems/Problem 2"
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\to \sum \sum 4x_ix_j\geq 0,</cmath> | \to \sum \sum 4x_ix_j\geq 0,</cmath> | ||
therefore we have to prove that | therefore we have to prove that | ||
− | <cmath>\sum \sum a_ia_j\geq 0</cmath> for every list | + | <cmath>\sum \sum a_ia_j\geq 0</cmath> for every list <math>x_i</math>, |
and we can describe this to | and we can describe this to | ||
<cmath>\sum \sum a_ia_j=\sum a_i^2 + \sum\sum a_ia_j(i\neq j)</cmath> | <cmath>\sum \sum a_ia_j=\sum a_i^2 + \sum\sum a_ia_j(i\neq j)</cmath> |
Latest revision as of 05:11, 24 April 2024
Contents
[hide]Problem
Show that the inequality holds for all real numbers .
Solution
then, since then, therefore we have to prove that for every list , and we can describe this to we know that therefore, --Mathhyhyhy 13:29, 6 June 2023 (EST)
Video solutions
https://youtu.be/cI9p-Z4-Sc8 [Video contains solutions to all day 1 problems]
https://youtu.be/akJOPrh5sqg [uses integral]
https://www.youtube.com/watch?v=P9Ge8HAf6xk
See also
2021 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |