How can I solve prove that $8(1-a)(1-b)(1-c)\le abc$ with the conditions below?
47059
How to establish this inequality: $(1-a)(1-b)(1-c) \geq 8abc$ for $a+b+c=1$?
130281
Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
105164
If $abc=1$, prove that $\frac{2}{(a+1)^2+b^2+1} + \frac{2}{(b+1)^2+c^2+1} + \frac{2}{(c+1)^2+a^2+1} \le 1$.
19909
if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$
98855
Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$
90670
Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.
157784
Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$.
31104
$a+b+c =0$; $a^2+b^2+c^2=1$. Prove that: $a^2 b^2 c^2 \le \frac{1}{54}$
50296
Prove $(-a+b+c)(a-b+c)(a+b-c) \leq abc$, where $a, b$ and $c$ are positive real numbers
121156
Prove that: $\gcd[a,b,c]=\frac{abc.\operatorname{lcm}(a,b,c)}{\operatorname{lcm}(a,b)\operatorname{lcm}(a,c)\operatorname{lcm}(b,c)}$
77064
Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$
81775
Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$
90438
To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$
78736
Check my proof that $(ab)^{-1} = b^{-1} a^{-1}$
43478
Prove that: $ m_a + m_b + m_c \le 4R + r $
109365
Prove that 1/2 + 1/4 + 1/8 ....... = 1
6970
If $abc=1$ and $a,b,c$ are positive real numbers, prove that ${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1$.