Homogeneous

A function $f(a_1,a_2,\ldots,a_n)$ 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

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)

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