Prove that if $a, b, c \geq 0$ then $(a + b)(b + c)( a + c) \geq 8abc$
52618
AM-GM inequality basic
112969
Show that $ab + bc + ac \ge a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab}$
11168
Proof of divisibility: if $a|b$ and $a|(b+c)$ then $a|c$
21726
Show that if $(b,c)=1$, then $(a,bc)=(a,b)(a,c)$.
50614
If $4\mid a+bc$ and $6\mid b+ac$ prove that $2\mid a^2-b^2$
39779
If $a,b,c>0$ and $a+b+c=1$, prove that $\sqrt{\frac{ab}{c+ab}}+\sqrt{\frac{bc}{a+bc}}+\sqrt{\frac{ac}{b+ac}} \le \frac{3}{2}$.
29935
If $a, b, c, d>0$ and $abcd=1$ prove that an inequality holds true
5369
Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
60122
Prove $(1+a+b)(1+b+c)(1+c+a)\ge 9(ab+bc+ca)$
89175
Proof of $(a+b)^{n+1}$
141102
If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$.
141369
If $a+b\equiv0\pmod m$ and $c+d\equiv 0\pmod m$ then $ac\equiv bd\pmod m$. Demonstration.
5410
Prove that $a^2 + b^2 \geq 8$ if $ x^4 + ax^3 + 2x^2 + bx + 1 = 0 $ has at least one real root.
112565
Prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{9}{a+b+c} : (a, b, c) > 0$
99692
Prove that $a^3+b^3+c^3 \geq a^2b+b^2c+c^2a$
90670
Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.
121177
Prove that $\frac{(2a+b+c)^2}{2a^2+(b+c)^2}+\frac{(2b+c+a)^2}{2b^2+(c+a)^2}+\frac{(2c+a+b)^2}{2c^2+(a+b)^2} \le 8$