Prove that for nonnegative \(a, b, c\) the following inequality holds: \[\left(\frac{a(b+c)}{4a+b+c} \right)^3+\left(\frac{b(c+a)}{4b+c+a} \right)^3+ \left(\frac{c(a+b)}{4c+a+b} \right)^3 \leq \frac{(a+b)(b+c)(c+a)}{72}\]
Tags: inequality
Prove that if \(a, b, c\) are sides of a triangle then \[\frac{(a^2+b^2+c^2)(a+b+c)}{9}\ge \frac{(-a^2+b^2+c^2)(a^2-b^2+c^2)(a^2+b^2-c^2)}{(-a+b+c)(a-b+c)(a+b-c)}.\]
Tags: inequality geometric inequality 3D
Prove that for reals \(a, b, c\ge 0\) \[(a^2-ab+b^2)(b^2-bc+c^2)(c^2-ca+a^2)\ge \frac{1}{8}(a^2+bc)(b^2+ca)(c^2+ab)\ge \frac{1}{9}(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2).\]
Tags: inequality
Find the largest number \(k>0\) with the property that for any \(a, b, c\ge 0\) the following inequality holds: \[(a-b)(b-c)(c-a)\le \frac{(a+b+c)^3}{k}.\]
Tags: inequality calculus