kyuhunl wrote:Is the median of the 3 different integers equal to their average (arithmetic mean)?

1) The median of the 3 integers is 19

2) The range of the 3 integers is 19

can anybody explain this for me?

Sure, kyuhunl! (Nice problem, by the way.)

$$a < b < c\,\,{\rm{ints}}$$
$$b\,\,\mathop = \limits^? \,\,{{a + b + c} \over 3}$$

$$\left( 1 \right)\,\,b = 19\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {18,19,20} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {18,19,21} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$

$$\left( 2 \right)\,\,c - a = 19\,\,\,\, \Rightarrow \,\,\,\,\left( {a,b,c} \right) = \left( {a,b,a + 19} \right)$$
$$b\,\,\mathop = \limits^? \,\,{{2a + b + 19} \over 3}\,\,\,\, \Leftrightarrow \,\,\,\,2b\,\,\mathop = \limits^? \,\,2a + 19\,\,\,\, \Leftrightarrow \,\,\,\,b\,\,\mathop = \limits^? \,\,a + {{19} \over 2}\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,$$
$$\left( * \right)\,\,\,a\,\,{\mathop{\rm int}} \,\,\,\, \Rightarrow \,\,\,\,\,a + {{19} \over 2}\,\, \ne {\mathop{\rm int}} \,\,\,\, \Rightarrow \,\,\,\,\,b \ne a + {{19} \over 2}\,\,\,\,\,\,\left( {b\,\,{\mathop{\rm int}} } \right)$$


The correct answer is therefore (B).


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

Regards,
Fabio.