Difference between revisions of "Homogeneous"
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| − | A | + | A [[polynomial]] <math>f(a_1,a_2,\ldots,a_n)</math> is said to be '''homogeneous''' if all its terms are of the same degree in <math>a_i</math>. |
This concept of homogeneity is often used in [[inequalities]] so that one can "scale" the terms (this is possible because <math>f(ta_1,ta_2,\ldots,ta_n)=t^kf(a_1,a_2,\ldots,a_n)</math> for some fixed <math>k</math>), and assume things like the sum of the involved variables is <math>1</math>, for things like [[Jensen's Inequality]] | This concept of homogeneity is often used in [[inequalities]] so that one can "scale" the terms (this is possible because <math>f(ta_1,ta_2,\ldots,ta_n)=t^kf(a_1,a_2,\ldots,a_n)</math> for some fixed <math>k</math>), and assume things like the sum of the involved variables is <math>1</math>, for things like [[Jensen's Inequality]] | ||
| − | ==Introductory== | + | == Problems == |
| − | == Intermediate== | + | |
| − | ==Olympiad== | + | === Introductory === |
| + | |||
| + | === Intermediate === | ||
| + | |||
| + | === Olympiad === | ||
*Let <math>a,b,c</math> be positive real numbers. Prove that | *Let <math>a,b,c</math> be positive real numbers. Prove that | ||
<math>\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1</math> ([[2001 IMO Problems/Problem 2|Source]]) | <math>\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1</math> ([[2001 IMO Problems/Problem 2|Source]]) | ||
| + | |||
| + | == See Also == | ||
| + | * [[Inequalities]] | ||
{{stub}} | {{stub}} | ||
| − | + | [[Category:Algebra]] | |
| + | [[Category:Inequalities]] | ||
[[Category:Definition]] | [[Category:Definition]] | ||
| − | |||
Latest revision as of 12:38, 14 July 2021
A polynomial 
is said to be homogeneous if all its terms are of the same degree in 
.
This concept of homogeneity is often used in inequalities so that one can "scale" the terms (this is possible because 
for some fixed 
), and assume things like the sum of the involved variables is 
, for things like Jensen's Inequality
Problems
Introductory
Intermediate
Olympiad
- Let
be positive real numbers. Prove that
be positive real numbers. Prove that
(Source)
See Also
This article is a stub. Help us out by expanding it.
