Prove that $a^ab^bc^c\ge (abc)^{(a+b+c)/{3}}$
90670
Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.
96327
Show that $a^x+b^x+c^x>(a+b+c)^x$ for all $a,b,c>0$ and $0<x<1$
112676
Inequality $(a+b+c)(ab+bc+ca)(a^3+b^3+c^3)\le (a^2+b^2+c^2)^3$
154716
Prove $\frac{1}{a^3+b^3+abc}+\frac{1}{a^3+c^3+abc}+\frac{1}{b^3+c^3+abc} \leq \frac{1}{abc}$
118507
Prove $\frac{ab}{1+c^2}+\frac{bc}{1+a^2}+\frac{ca}{1+b^2}\le\frac{3}{4}$ if $a^2+b^2+c^2=1$
56101
Prove the inequality $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$
138216
$a,b,c$ are positive real numbers such that, $a+b+c\ge abc$. Prove that $a^2+b^2+c^2\ge \sqrt{3}abc$
110944
Proving that $AB-BA=cI$ for nontrivial $c \in \mathbb{C}$
120954
Prove that: $\sqrt[3]{a_1^3+ a_2^3 +\cdots+a_n^3} \le \sqrt{a_1^2 + a_2^2 +\cdots+a_n^2}$
60122
Prove $(1+a+b)(1+b+c)(1+c+a)\ge 9(ab+bc+ca)$
90902
Prove $\frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$.
37536
Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$.
72470
Prove that $2|ab| \leq a^2 + b^2$ and $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$
7015
Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian
130281
Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
19940
How to show that $A^3+B^3+C^3 - 3ABC = (A+B+C)(A+B\omega+C\omega^2)(A+B\omega^2+C\omega)$ indirectly?
56311
Proving $(1+a_{2})^{2}(1+a_{3})^{3}\cdots(1+a_{n})^{n}\ge n^n$ for positive reals $a_2,\ldots, a_n$ whose product is $1$