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.

_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)

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