Difference between revisions of "Homogeneous"
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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 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>. | ||
| − | 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]] |
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| + | ==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]]) | ||
{{stub}} | {{stub}} | ||
Revision as of 01:19, 11 May 2019
A function 
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
Introductory
Intermediate
Olympiad
- Let
be positive real numbers. Prove that
be positive real numbers. Prove that
(Source)
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