Is abc at least 4?

Hi, Mitch! Thanks for the words and for your beautiful contribution!

\(abc\,\,\mathop \ge \limits^? \,\,4\)

\(\left( 1 \right)\,\,b + c\,\, \ge 2\,\,\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr \\
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {4,1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.\)


\(\left( 2 \right)\,\,ab \ge ac \ge 4\,\,\,\,\,\left\{ \matrix{\\
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {4,1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \\
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( { - 2, - 2, - 2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\)


\(\left( {1 + 2} \right)\,\,a \ne 0\,\,\,\,\,\left( {ac \ne 0} \right)\,\,\,\,::\,\,\,\,\left\{ \matrix{\\
\,a < 0\,\,\,\, \Rightarrow \,\,\,\,b < 0\,\,\,\,\left( {ab > 0} \right)\,\,\,\,\,and\,\,\,\,\,c < 0\,\,\,\left( {ac > 0} \right)\,\,\,\,\mathop \Rightarrow \limits^{\left( 1 \right)} \,\,\,\,{\rm{impossible}} \hfill \cr \\
\,a > 0\,\,\,\, \Rightarrow \,\,\,\,b > 0\,\,\,\,\left( {ab > 0} \right)\,\,\,\,\,and\,\,\,\,\,c > 0\,\,\,\left( {ac > 0} \right) \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,a,b,c\,\,\, > 0\,\,\,\,\,\left( * \right)\)

\(\left. \matrix{\\
ab \ge 4\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,c\,\,\left( * \right)} \,\,\,abc \ge 4c\,\,\, \hfill \cr \\
ac \ge 4\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,b\,\,\left( * \right)} \,\,\,abc \ge 4b \hfill \cr} \right\}\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,\,\,2abc \ge 4\left( {b + c} \right)\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{:\,2} \,\,\,\,\,\,\,abc \ge 2\left( {b + c} \right)\,\,\,\,\mathop \Rightarrow \limits^{\left( 1 \right)} \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle\)


The correct answer is therefore (C).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.