Prove that 6 | (a+b+c) if and only if 6 | (a^3 + b^3 +c^3).
29963
$6$ divides $a+b+c$ if and only if $6$ divides $a^{3}+b^{3}+c^{3}$
73029
If a,b,c are sides of a triangle, prove: $ \sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \le \sqrt{a} + \sqrt{b} + \sqrt{c} $
11168
Proof of divisibility: if $a|b$ and $a|(b+c)$ then $a|c$
109133
$a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$
136854
Prove: For a,b,c positive integers, ac divides bc if and only if a divides b
148744
Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \ge \sqrt{(bc)^a(ac)^b(ab)^c}$
104124
Prove or disprove: $A-(B-C) = (A-B)-C$
154716
Prove $\frac{1}{a^3+b^3+abc}+\frac{1}{a^3+c^3+abc}+\frac{1}{b^3+c^3+abc} \leq \frac{1}{abc}$
23869
Prove the implication "if $ac>bc$ and $c>0$, then $a>b$" using the contrapositive
138894
Prove that if $3\mid(a^2+b^2)$,then $3\mid a$ and $ 3\mid b$
42545
If $ a,b,c\in \left(0,\frac{\pi}{2}\right)\;,$ Then prove that $\frac{\sin (a+b+c)}{\sin a+\sin b+\sin c}<1$
52204
Proof of one inequality $a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$
136367
Prove $ ∀a, b, c ∈ Z,$ if $ab + ac ≡ 3$ (mod 6) then b $\not\equiv$ c (mod 6).
16866
How to Prove $a^3 \cos (B-C) + b^3 \cos (C-A)+ c^3 \cos (A-B)=3abc$?
96327
Show that $a^x+b^x+c^x>(a+b+c)^x$ for all $a,b,c>0$ and $0<x<1$
136868
Prove that $\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$ if $(a+b+c)^2(a^2+b^2+c^2)=27$
15116
Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$