Difference between revisions of "Cauchy-Schwarz Inequality"
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| − | The Cauchy-Schwarz [[Inequalities | Inequality]]states that, for two sets of real numbers <math>a_1,a_2,\ldots,a_n</math> and <math>b_1,b_2,\ldots,b_n</math>, the following inequality is always true: | + | The Cauchy-Schwarz [[Inequalities | Inequality]] states that, for two sets of real numbers <math>a_1,a_2,\ldots,a_n</math> and <math>b_1,b_2,\ldots,b_n</math>, the following inequality is always true: |
<math>\displaystyle({a_1}^2+{a_2}^2+...+{a_n}^2)({b_1}^2+{b_2}^2+...+{b_n}^2)\geq(a_1b_1+a_2b_2+...+a_nb_n)^2</math> | <math>\displaystyle({a_1}^2+{a_2}^2+...+{a_n}^2)({b_1}^2+{b_2}^2+...+{b_n}^2)\geq(a_1b_1+a_2b_2+...+a_nb_n)^2</math> | ||
This inequality is used very frequently to solve Olympiad-level Inequality problems, such as those on the [[United States of America Mathematics Olympiad | USAMO]] and [[International Mathematics Olympiad | IMO]]. | This inequality is used very frequently to solve Olympiad-level Inequality problems, such as those on the [[United States of America Mathematics Olympiad | USAMO]] and [[International Mathematics Olympiad | IMO]]. | ||
Revision as of 23:02, 17 June 2006
The Cauchy-Schwarz Inequality states that, for two sets of real numbers 
and 
, the following inequality is always true:
This inequality is used very frequently to solve Olympiad-level Inequality problems, such as those on the USAMO and IMO.
