For $a,b,c>0$ prove that $abc(a+b+c) \le a^3 b + b^3 c + c^3 a$
117281
How prove this inequality $x^3y+y^3z+z^3x\ge xyz(x+y+z)$
112969
Show that $ab + bc + ac \ge a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab}$
104498
Show that $x^3y^3(x^3+y^3) \leq 2$
23080
Prove that $a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc.$
6970
If $abc=1$ and $a,b,c$ are positive real numbers, prove that ${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1$.
15116
Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$
19909
if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$
26741
How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
60673
To prove $\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$
82612
Show that $\frac c {1+c} \le \frac a {1+a} + \frac b {1+b}$ , for $c \le a+b$ and $a,b,c \ge 0$
120954
Prove that: $\sqrt[3]{a_1^3+ a_2^3 +\cdots+a_n^3} \le \sqrt{a_1^2 + a_2^2 +\cdots+a_n^2}$
48484
Prove that $6|a+b+c$ if and only if $6|a^3+b^3+c^3$
90902
Prove $\frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$.
14563
Show $\frac{b}{c} + \frac{c}{a} + \frac{a}{b} \ge 3$ for $a,b,c > 0$.
12954
Prove if $ac>bc$ and $c>0$, then $a>b$
109133
$a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$
141102
If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$.