If a+b+c=1 and (a,b,c>0) then

	Column I		Column II
A)	abc	P)	$\le$64
B)	$\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}$	q)	$\le$8/27
C)	$\left(1+\frac{1}{\mathrm{a}}\right)\left(1+\frac{1}{\mathrm{b}}\right)\left(1+\frac{1}{\mathrm{c}}\right)$	R)	$\ge$9
D)	(1 - a)(1 - b)(1 - c)	S)	$\le$1/27

	A	B	C	D
1)	S	R	Q	P
2)	R	S	P	Q
3)	S	R	P	Q
4)	R	S	Q	P

$$\begin{gathered} \mathrm{A}) \mathrm{A}.\mathrm{M}\ge \mathrm{G}\cdot \mathrm{M}\Rightarrow \frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{3}\ge (\mathrm{abc}{)}^{1/3}\Rightarrow \mathrm{abc}\le \frac{1}{27} \\ \text{ B) }\mathrm{A}\cdot \mathrm{M}\ge \mathrm{H}\cdot \mathrm{M}\Rightarrow \frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{3}\ge \frac{3}{\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}}\Rightarrow \frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}\ge 9 \\ \text{ C) }\frac{\left(1+\frac{1}{\mathrm{a}}\right)+\left(1+\frac{1}{\mathrm{b}}\right)+\left(1+\frac{1}{\mathrm{c}}\right)}{3}\ge {\left(\left(1+\frac{1}{\mathrm{a}}\right)\left(1+\frac{1}{\mathrm{b}}\right)\left(1+\frac{1}{\mathrm{c}}\right)\right)}^{1/3} \\ \begin{array}{r}\begin{array}{rl} & 1+\frac{1}{3}\left(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}\right)\ge {\left(\left(1+\frac{1}{\mathrm{a}}\right)\left(1+\frac{1}{\mathrm{b}}\right)\left(1+\frac{1}{\mathrm{c}}\right)\right)}^{1/3}\\ & \left(1+\frac{1}{\mathrm{a}}\right)\left(1+\frac{1}{\mathrm{b}}\right)\left(1+\frac{1}{\mathrm{c}}\right)\le {\left(1+\frac{1}{3}\left(9\right)\right)}^3\le 64\end{array}\\ \begin{array}{rl} & \text{ D) }\frac{(1-\mathrm{a})+(1-\mathrm{b})+(1-\mathrm{c})}{3}\ge \left[\right(1-\mathrm{a}\left)\right(1-\mathrm{b}\left)\right(1-\mathrm{c}){]}^{1/3}\\ & \frac{3-1}{3}\ge \left[\right(1-\mathrm{a}\left)\right(1-\mathrm{b}\left)\right(1-\mathrm{c}){]}^{1/3}\\ & (1-\mathrm{a})(1-\mathrm{b})(1-\mathrm{c})\le \frac{8}{27}\end{array}\end{array} \end{gathered}$$