ashwink wrote:
If m, s are the average and standard deviation of integers a, b, c, and d, is s > 0?
1. m > a
2. a + b + c + d = 0
\(a,b,c,d\,\,\,\,{\text{ints}}\)
\(m = \frac{{a + b + c + d}}{4}\)
\(s = \sigma \left( {a,b,c,d} \right)\)
\(s\,\,\mathop > \limits^? \,\,0\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\boxed{?\,\,\,:\,\,\,\,a = b = c = d\,\,\,\,{\text{false}}}\,\)
\(\left( 1 \right)\,\,\,m > a\,\,\,\, \Rightarrow \,\,\,\,a = b = c = d\,\,\,\underline {{\text{false}}} \,\,\,({\text{otherwise}}\,\,m = a)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle\)
\(\left( 2 \right)\,\,\,\left\{ \begin{gathered}\\
\,{\text{Take}}\,\,\left( {a,b,c,d} \right) = \left( {0,0,0,0} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\, \hfill \\\\
\,{\text{Take}}\,\,\left( {a,b,c,d} \right) = \left( {1, - 1,0,0} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\, \hfill \\ \\
\end{gathered} \right.\)
The above follows the notations and rationale taught in the GMATH method.