Proving $a^ab^bc^c\ge(abc)^{(a+b+c)/3}$ for positive real numbers.
90670
Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.
2019
Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
37536
Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$.
44705
Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$
17100
Prove that $\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}+\frac{a^2+b^2}{a+b} \ge a+b+c$
50296
Prove $(-a+b+c)(a-b+c)(a+b-c) \leq abc$, where $a, b$ and $c$ are positive real numbers
101362
Let a,b,c be positive real number, proof.
124868
Prove that $\frac{bc}{b+c}+\frac{ca}{c+a}+\frac{ab}{a+b}<\frac{a+b+c}{2}$
56101
Prove the inequality $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$
112969
Show that $ab + bc + ac \ge a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab}$
95643
Proving $\sum_{cyc} \sqrt{a^2+ab+b^2}\geq\sqrt 3$ when $a+b+c=3$
111267
Prove that: $2^a+3^b<3a+4b$
111847
For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$
31173
How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers?
15599
Given that a,b,c are distinct positive real numbers, prove that (a + b +c)( 1/a + 1/b + 1/c)>9
89175
Proof of $(a+b)^{n+1}$
99692
Prove that $a^3+b^3+c^3 \geq a^2b+b^2c+c^2a$