Difference between revisions of "Homogeneous"

Latest revision as of 12:38, 14 July 2021

A polynomial $f(a_1,a_2,\ldots,a_n)$ is said to be homogeneous if all its terms are of the same degree in $a_i$ .

This concept of homogeneity is often used in inequalities so that one can "scale" the terms (this is possible because $f(ta_1,ta_2,\ldots,ta_n)=t^kf(a_1,a_2,\ldots,a_n)$ for some fixed $k$ ), and assume things like the sum of the involved variables is $1$ , for things like Jensen's Inequality

Problems

Introductory

Intermediate

Olympiad

Let $a,b,c$ be positive real numbers. Prove that

$\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1$ (Source)

@@ Line 1: / Line 1: @@
-A function <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>.
+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>.
+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]]
+== Problems ==
+=== Introductory ===
+=== Intermediate ===
+=== Olympiad ===
+*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]])
+== See Also ==
+* [[Inequalities]]
 {{stub}}
+[[Category:Algebra]]
+[[Category:Inequalities]]
 [[Category:Definition]]
-[[Category:Elementary algebra]].

Page

Toolbox

Search