BTGmoderatorDC wrote:4 raffle 'winners' were each given the opportunity to select one envelope from a group of 4 envelopes and won the amount of money that was written inside the envelope. After each winner selected an envelope and was awarded a prize, a new set of 4 envelopes was offered to the next winner, and so on. If the individual amounts in the envelopes were $1,000, $2,000, $3,000 and $4,000, did any of the winners receive the largest prize?

(1) No two winners received the same prize.
(2) The total prize money awarded was $10,000.

The numbers related to amounts are presented in thousand dollars.
\[?\,\,\,:\,\,\,\$ 4\,\,{\text{for}}\,\,{\text{someone}}\]
(1) Sufficient:

There were 4 people, each one had chosen one different value from 4 possible values. Each value was chosen once ($4 included).


(2) Insufficient: (A, B, C, D are the "winners")
\[\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {A,B,C,D} \right)\, = \left( {1,2,3,4} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,\left( {A,B,C,D} \right)\, = \left( {1,3,3,3} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\,\left( {A + B + C + D = 10} \right)\]

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.