Question

If $a,b,c\in R^{+}$ and $a^{2}bc+4abc+4bc=81$ such that ($2a+b+c+4$) assumes its least value, then

• A. $a$ is a prime number
• B. $a+b+c$ is a prime number
• C. $b+c-a$ is a prime number
• D. $a^3+b^3+c^3$ is a product of two prime numbers

Answer 1

Answer: (B), (C), (D)

$a^3+b^3+c^3$ is a product of two prime numbers

$bc(a^2+4a+4)=81$
$\Rightarrow bc(a+2{)}^2=81$

AM $\ge$ GM
$\frac{(a+2)+(a+2)+b+c}{4}\ge \left[\right(a+2{)}^{2}bc{]}^{\frac{1}{4}}$
$2a+b+c+4\ge 12$
equality holds when $a+2=b=c=3$
$\Rightarrow a=1,b=3,c=3$