<p>
</p>
{a, b, c, d, e}
For the list of numbers above, there is exactly one mode. Is the range of the list greater than the mode of the list?
(1) At least one of the numbers in the list is negative.
(2) At least one of the numbers in the list is zero.
\[{\text{WLOG}}\left( * \right)\,\,\,\,a \leqslant b \leqslant c \leqslant d \leqslant e\,\,\,\,\,\,\left( {{\text{single}}\,\,{\text{Mo}}} \right)\]
\[{\text{?}}\,\,\,{\text{:}}\,\,\,{\text{Mo}}\,\,\mathop < \limits^? \,\,e - a\]
\[\left( 1 \right)\,\,\,\,a < 0\,\,\,\, \Rightarrow \,\,\,\,e - a = e + \left( { - a} \right) > e\]
\[?\,\,\,:\,\,\,{\text{Mo}} \leqslant e < e - a\,\,\,\,\, \Rightarrow \,\,\,\,{\text{Mo}} < e - a\,\,\,\,\,\,\left( {{\text{Suf}}{\text{.}}} \right)\]
\[\left( 2 \right)\,\,\,{\text{Insuf}}{\text{.}}\,\,\,\left\{ \begin{gathered}
\,\,\,\left\{ {0,0,0,0,1} \right\}\,\,\,\, \Rightarrow \,\,\,\,{\text{Yes}}\,\,\,\left( {{\text{Mo}} = 0\,\,;\,\,1 = e - a} \right) \hfill \\
\,\,\,\left\{ {0,1,1,1,1} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{No}}\,\,\,\,\,\left( {{\text{Mo}} = 1\,\,;\,\,1 = e - a} \right) \hfill \\
\end{gathered} \right.\]
(*) WLOG = Without loss of generality
POST-MORTEM: {0,0,1,1,2} is an example of a list that does not have a single mode (0 and 1 are two distinct modes of this list).
The solution above follows the notations and rationale taught in the <b>GMATH</b> method.
